5 Academy's 2022-2023 ACT® Math Practice Test FULL Walkthrough + ACT® Math Strategies

hey guys it's shahir from five academy welcome back to another five academy video so today we're going to be going through a full 60 question 60 minute act math test i'm obviously going to take a little longer than 60 minutes so i can describe each problem to you and really break it down so you understand what's happening but the question is why are we going to do this why are we going to go through all of these problems well the answer is because the more problems you go through the more tests you take the more content you review on the exam especially on the math and english tests the more skills you'll learn and the more familiar you'll become with the content that's asked about on every single one of these practice tests it's the same skills again and again and you can learn them at least to an extent through your practice tests now if you want to learn these uh in a bit more advanced way uh more than just taking practice tests and watching videos like this we have on our website actually a resource where we sort all these practice tests by skill so you can uh input your practice test results uh essentially take the test on our system and we will give you data and insights on which skills you're doing better or worse on so if you want to check that out that's on our website uh you can get access to that um and the very last thing i'll say before we get started uh you can also get more of these tests for free as well as the same p test pdf so you can take the test on your own do it you know simulate it just how you'd want to so i recommend you do that before you watch this video so that you can take the test correct it grade it there's a score chart at the bottom of this pdf that i'm looking at you can do that and then come back to this video at the end and essentially watch the parts of the video for any problems you didn't really understand or that you want to learn and understand a bit better so i'd advise that you do that and make the most of that and we won't go super super super in depth we won't spend more than two three minutes on any given problem here for the most part so if you want more in-depth analysis for any of these skills because there are you know about 30 40 50 skills here in this video then definitely check out our website we have lots of resources tutoring programs free and paid that help students and you can actually schedule free tutoring with me to go over some of these problems if you want i will link free tutoring sessions with me and a scheduler for that down below it takes literally 30 seconds to a minute to set that up all right so without further ado let's get started first problem if 6 times x minus 7 parentheses equals 4 then what does x equal so we can do this by distributing the 6 to the x and the seven so let's distribute that out six times x minus six times seven which is 42 is equal to four let's add 42 on both sides 6x equals 46 that means if we divide both sides by six we get x equals 46 over six we divide both top and bottom by two we get 23 over three the daily fees for camping at this campground are 15 per day and five dollars per person last summer daily fees were paid for d days and p persons which of the following expressions gives the total amount collected for daily fees last summer so let's just say we were going for three days seven people how much would we pay fifteen dollars times three days right and then we'd pay five dollars per seven persons that means we're paying 45 plus 35 equals 80. right that's our total fee now in this case what did i really just do here how can i get a function out of this well we're told that we're pricing for d days and p persons right that's what it says right here so instead of plugging in a number for d or p i can just plug in d or p itself and just make that a variable so your answer for two should be b a function f of x is defined as f of x equals negative four times x to the third which of the following is f at 3. so you're just going to plug in 3 for x negative 4 times 3 to the third which is going to give you negative 4 times 27 which is just 3 to the third and the negative 108 is going to be your answer if you plug it into the calculator a rectangle has an area of 48 feet squared and a perimeter of 32 feet what is the length of the longest side of the rectangle so if i draw a rectangle i have length width length width the area is going to be length times width okay the perimeter is going to be the sum of all the side lengths so in this case that's just going to be l plus l plus w plus w which is 2l plus 2w and i know that my area is 48 feet squared my perimeter is 32 feet okay so what i have here is two functions and two unknowns i can create a system of equations to solve so the way i'm going to do that let's move this make it a bit smaller that's good okay i'm going to take this first equation right here and solve it for l okay l is going to be 48 divided by w if you just divide both sides by w then i'm going to plug this l in here so i get 2 times 48 over w plus 2w equals 32 that becomes 96 over w plus 2w equals 32 and then i'll multiply each term by w because i want to solve for w you'll notice that if i multiply everything by w this turns into a quadratic 96 plus 2w squared plus 32w or sorry equals 32w now i can move everything to one side 2w squared minus 32w plus 96 equals zero okay i'm going to quickly delete what's up here move this up now i'm going to divide everything by two so we get w squared minus 16 w plus 48 equals zero now i can factor this or do the quadratic formula i like to factor it because i i see that there's factors of 48 12 and 4 or negative 12 negative 4 that equal negative 16 when you add them together so w minus 12 w minus four equals zero that means that width is equal to is it going to be either 12 or four um really that just means length or width are going to be 12 or four so your length is going to be 12 your width is going to be four that means your answer here is d which is 12. okay i mean if you multiply these two together you get 48. if you plug these into the formula 2l plus 2w essentially 2 times 12 plus 2 times 4 you end up getting the answer that i said here for two consecutive integers the sum of double the smaller and triple the larger is 63 what is the sum of the two integers so two consecutive integers let's just say for example it's 33 and 34 right you can make a system of equations out of this so let's just say the first one is x the second one is y and i know that y is going to be equal to x plus one or literally i could say the first integer is x and then the second one is x plus one okay and then i can go through and solve this so we're told two consecutive integers the sum of double the smaller a number so the smaller number is going to be x right so let's just kind of convert this statement into words so the sum we're going to be adding two things together and then it says later on is 63 okay the sum of double the smaller number double the smaller number so the smaller number is x double the smaller numbers gonna be two times x okay the sum of this and this so that means we're this is just the first term now the second term is going to be what comes after and triple the larger so the larger integer is what x plus one and it says triple which means we're multiplying bit by three okay so the sum let's just erase this kind of clean it up walk through it again the sum plus of double the smaller 2x and triple the larger is equals 63 63 okay so we have all the terms there all we need to do now is solve it so 2x plus 3 times x plus 1 equals 63. i can distribute the 3 out so we get 2x plus 3x plus 3 equals 63. i can combine these x terms and then i can also move 3 to the other side by subtracting 3 on both sides 5x equals 60 divide both sides by 5 x equals 12. okay now the question doesn't ask just for what the smaller number is that's why you don't see 12 as an answer option here the question is asking for what's the sum of the smaller and the larger so what essentially is x plus x plus 1. so i know that x is equal to 12 that means that this is obviously going to be 13 right if i add those together you get 25 and that's your answer given a equals bc squared and a equals this b equals this which of the following is c okay let's just plug in what we know and see what we end up with it looks like we have three unknowns and uh three three variables and two of them are known so we can solve for the third 250 equals 10 times c squared so i can immediately solve for c just divide both sides by 10 and then take the square root so you get plus or minus five the only answer option here that we see is five the first term in a sequence geometric sequence is three the rest of the sequence continues as is shown below which of the following is the eighth term in the sequence the best way to do this is to just continue the pattern and to notice the pattern so it looks like here i'm doing times negative two times negative two times negative two so we're going to continue that what's negative 424 times negative 2 you get 48 and then i get negative 96 then i get 192 then i get negative 384 7 should be b and which of the following ways are the fractions in arranged in increasing order the best way to do this is to take each of these fractions and convert it to decimal form so 2 over 3 is equal to 0.6 repeating essentially you could think of it this way 3 over 5 is equal to 0.6 just flat 0.60 and then we have 5 over 8 which is equal to 0.625 now what i can do is just i can just sort these in decimal form because it's obvious which one's larger than the other clearly 2 over 3 is the biggest so i'm going to put this here and then this one's these two are already arranged as is needed okay so from here left to right these increase and this right here gives me my order all i have to do is sort these and assort these accordingly and i have my order eight should be b which of the following is the side length in feet of the isosceles right triangle shown below given that the hypotenuse is six feet long so the hypotenuse is always the longest side it's also the side that's opposite the right angle okay now the side length in feet of the uh so when it's a side length it's asking for the leg length the side leg length okay so we know that these have to be the same because it's isosceles isosceles means that there are two side lengths that are the same clearly this is this can't be the one that's the same that has another that's the same dimension because this is the hypotenuse it's the longest side okay so we know that these two are the same also since it's an isosceles right triangle we know that both of these angles are 45 degrees so we have a couple different ways we can solve this one of them is to use a sine cosine tangent ratio so i could say sine of 45 is equal to opposite over hypotenuse so sine of 45 is equal to x over 6. i can just solve for x x equals 6 sine 45 you can do this on your calculator you'll get the decimal equivalent of 3 squared of 2 i believe so if you do sine of 45 you multiply by 6 you'll get 3 square root of 2 just to confirm okay you can now now you don't just want a decimal equivalent if you get a decimal equivalent and you're not sure which answer option to choose because none of them are in decimal form you could just plug in each answer option and find the decimal equivalence of each and then just match your answer here otherwise if you remember that sine 45 is equal to square root of 2 over 2 then 6 squared of 2 over 2 is going to give you 3 square root of 2. okay so that that's that's another easy way to solve it the the fastest way to solve it is to do this so you need to keep in mind that this these ratios are true for any isosceles right triangle if the side lengths are labeled as x that means that the uh hypotenuse is going to be that same x multiplied by square root of two alternatively so you see to go from here to here you multiply by square root of two okay or to go from here to here you divide by square root of two okay now since i already know this all i have to do is divide by square root of two or essentially take six which is uh 3 times 2 and then divided by square root of 2. 2 divided by square root of 2 is just going to be square root of 2 so i end up getting 3 square root of 2. okay that's another way to do it again don't confuse yourself too much i'm just giving you different options if any one of those made sense then stick with that one method in the coordinate plane which of the following is the slope of this line so slope is given by this form where m is your slope so we need to basically just solve this for y i can start by subtracting 2 on both sides so we end up getting uh 3x minus 2 equals 5y let's write it by flipping right and left so you might think okay mx plus b we're good right but the thing is y has to be by itself i need to get rid of this five okay let's do just that by dividing five on both sides or essentially all the terms y equals three over five x minus two over five now that right there is your slope so 10 is going to be b which of the following is a factored form of the expression the way you can factor this is with the box method the way you do the box method is you write the first term in the top left of your 2x2 box you write the last term the c term in the bottom right and then what you do is you multiply that coefficient of the first term with the constant in the last term and you end up getting negative 24. this is what you do when you have negative 24. you think of factors of negative 24 that add or subtract to the middle number to the b term to negative two okay so what are the factors of negative 24 we have 124 that pair doesn't work 2 12 that pair doesn't work 3 8 that pair doesn't work 4 6. so we can get negative 4 6 or negative 6 4 and these two will actually add up to negative 2 okay now uh i'm gonna have to choose this pair because the when i add negative 6 plus 4 i get negative 2. but if i add 4 plus negative sorry if i do 6 plus negative 4 i get positive 2. i don't want positive 2 i want negative 2 all right so i'm going to take both of these negative 6 and 4 and i'm going to write them in their own box with an x term okay because these two sum to give me the middle term think of it that way all right so really what i just did is i expanded this over four boxes the three terms over four boxes by splitting the middle term into two factors that i know can uh multiply to give me the anc term product okay now what i do last step and if you've seen the factoring video you've seen how i do this so um what you do is you take each column and you find a common factor the greatest common factor of each column and you write them up here you do the same thing for each row you write the greatest common factors on the side okay so i'm gonna do that for the first two columns greatest common factor of negative 6x and negative eight is negative two and if uh what you do with the sign is you take the sign of whatever term you've written here okay so if this is negative i'm going to have negatives here and here if this is positive i'm going to positive there okay so since that's a negative 6x i'm going to have a negative 2. okay greatest common factor here is x greatest common factor here is 3 x you can include x in there because they both have it here the greatest common factor is going to be four so your final answer is going to be x minus two times three x plus four and just to check make sure we did it correctly what you can do is just foil it out so you do 3x squared plus 4x minus 6x minus 8. and that does check out it gives you the same thing it's actually exactly what we have down here all right so we've factored that with the box method for more information on that check out the factoring video the skill video for factoring the sum of the real numbers x and y is 10 the difference is 8. what is the value of x y so we know the sum is 10 x plus y have to add up to 10. it says their difference is 8 meaning that 1 minus the other is 8. the thing is it doesn't matter how you order the one minus the other because as long as the sum is 10 it doesn't really matter what you get for x and y individually because again you're just going to multiply them at the end so again it doesn't matter which one is which it just matters that you get the right values okay so i'm going to just stick with doing it this way because it's just a bit more logical now i have a system of two equations and i have two unknowns so i can use substitution or just actually use elimination let's just do that just add both these equations to each other and i'll get one of the variables so 2x plus 0 because these y's will cancel when i add these two equations together will give me 18. if i add 10 and 8. 2x equals 18 that means x is equal to 9. okay now if i remember that x minus y is equal to 8 that means 9 minus y is equal to 8 and that means that y is equal to 1. so y is 1 x is 9 9 times 1 is just 9. okay which of the following is the least common factor least common multiple of these three numbers now one way you could do this is to just take each answer option if you want to cheat on the uh through the problem take each answer option and just divide it by all three of the numbers and see what's the smallest answer option that's divisible by all three numbers okay so for example i'm going to start with the smallest value let's do 240 divided by 36 do 240 divide by 40 240 divide by 48. and check do i get a whole number for all all of these 240 divided by 36 that does not give me a whole number so for that reason since it's not divisible by 36 240 is not actually a multiple of 36 okay so this is out you're gonna have the same issue with 500 720 will be the smallest one that's divisible by all of these okay now if you want a more holistic method a more in-depth method that really helps you understand the skill behind this you can make a factor tree of all three of these um numbers uh and then you take the factor trees of all three i'm not going to go in depth on this but you take the factories of all three you find all of the unique factors that are across all three of these factor prime factorizations and then you multiply those unique factors together there's more on that in the video but again for this type of problem you don't really need to know what you're doing you can just reverse solve it by checking each answer option like i showed you in abc the sum of the measures of a and b is 53 and note that this is a triangle which of the following is the measure of angle c so let me just draw a random triangle a b c this is definitely not drawn to scale so we know angles a and b these two angles add up to 53 together okay not each but together let's just label this one x okay so uh how can we use that to find angle c well we know that angle a plus angle b plus angle c has to equal 180 degrees because the interior angles of a triangle always add up to 180 degrees we know that angle a plus angle b already equal 53 so 53 plus angle c will give me 180 that means that angle c is equal to 180 minus 53 which is just going to be 20 127 all right so i really just only had one unknown in that problem we can quickly solve by solving for that unknown absolute value okay so what you do is you always start with the innermost operation so negative 7 plus 4 is equal to negative 3. so i'm going to rewrite this just with this simplification negative 5 absolutely negative 3. now what's the absolute value of negative 3 it's just positive 3. so negative 5 times positive 3 is going to give me negative 15. okay for the triangle shown below which of the following is the value of tan of theta if you remember your sohcahtoa acronym then this problem should be trivial so toa tan theta equals opposite over adjacent t o a right now opposite is the side that the angle opens up to b opposite over adjacent and the adjacent side is going to be the side that the angle is touching now you might think okay the angle is touching two sides it's touching c as well well c is the hypotenuse and how do you notice the hypotenuse it's because it is opposite of the 90 degree angle the largest angle so that means it's the largest sign if it's opposite the largest angle okay so opposite is here adjacent is sorry opposite adjacent b over a how do we simplify this expression so what i'm going to do is i can take this negative and distribute it to each of these terms and then just turn this all into addition okay so we're going to take 6a minus 9b plus 2c plus negative 3a plus 4 would be minus 4c essentially just flipping the signs of each of these terms okay now all we have to do is add combine like terms so we have a 6a and then we're subtracting a 3a so we have to have 3a in our answer okay we have a negative 9b and we're subtr we're adding a 4b that gives me negative 5b okay so we have to have a negative 5b in our answer that means that these two are out and then you'll see if you do 2c minus 4c you'll end up getting negative two c okay so the problem checks out seventeen should be c twelve percent of fifty is one sixth of which are the following numbers so twelve percent of fifty how do we find twelve percent of fifty what you do is you take the percent and you divide it by 100 and that gives you the decimal equivalent of that percent so 12 over 100 times 50 is that can be converted to an equal sign 1 6 you just literally write 1 over 6 of which are the following numbers some number okay i'm going to plug in x now all we have to do here is just solve for x as long as i can do all these operations or reverse some of the operations i should be able to get this answer very quickly okay so i can do this on my calculator let me actually before i do that let me get rid of this 1 over 6 so i can isolate x so the way you're going to get rid of 1 over 6 is just multiply both sides by 6. so your answer is going to be 6 times 12 over 100 times 50 is equal to x this gives you 6 times 12 times 50 divide by 100 you get 36 okay so that number x is 36. very simple it's only two operations really a circle in the standard coordinate plane passes through this point and has its center at this point line segment fg is a diameter of this circle which of the following are the coordinates of point f so let's go ahead and first plot the points we have we want to visualize this problem so that it becomes a little bit easier to interpret okay so we have passes through 5 5 easy to plot it's going to be even so let's label that as g and the center of the circle is at 3 comma 1 so it's going to be somewhere like this okay this is point e that's the center of the circle so your circle is going to look something like this make it a bit smaller there we go okay that's your circle now uh it says that f g is a diameter of the circle which means that it goes all the way across from g to f f is down here somewhere okay now we don't know the coordinates of f so how do we find the coordinates of f what i would recommend you do is think what do you have to do to go from g to e well you have to go you have to go down right and then you have to go left a bit so in order to go from e to f you can do the same thing because e is a midpoint that means that whatever you do from g to e you have to go from e to f to get literally e to f um it's the same exact length and distance of travel okay that's what a midpoint is so how do we know how far up and down we're going from g to e well if i look at the coordinates five comma five and three comma one i know that the x value is going left two right that's why it changes by two so we're going minus two the y value goes down by four so i'm gonna do the same thing from e to f down four left two so you can take your e point your midpoint and subtract two from the x coordinate and then subtract 4 from the y coordinate 1 comma negative 3. and this looks like 1 comma negative 3. it looks like we did it properly 19 should be e all right moving on alex doesn't know if he should buy a season pass ticket for his college football teams 18 home games this season the price of an individual ticket is this much and the cost of a season pass is this much what is the minimum number of home football games he must attend for the cost of a season pass to be less than the total cost of buying individual tickets for each go home game he watches okay so obviously if you're only going to see two or three games you'll pay like 30 40 in tickets and that means that the season pass is not worth it this question is really asking how many tickets would you have to buy how many individual tickets until you get to the to the 150 dollar mark or exceed the 150 mark where the 150 price point becomes more worth it so one way to do this is just list out all the multiples of 12 right see how many multiples it takes you to get to 150 you're going to get to 144 and then you're going to get to 156 this number is 12 times 13 so 13 tickets at 13 tickets you're going to be if you list this out you'll see that you are exceeding the season pass price so if you're going to go to 13 or more games that means that it's better to just buy the season pass okay the other way you can do this is just divide 50 150 by 12 tickets and see how much uh or no 12 or 12 per game per game right what this will do is will give you an answer of like i think 12.5 or something let me just check that 150 divided by 12 it should be 12.5 gives you 12.5 games okay at the original price so if you go to 12 and a half games using the original rate you'll hit 150 flat so anything more than 12 and a half games is going to be 13 games or more right that's where the 150 dollars becomes worth it right so you can use this to get that number 13. you obviously have to round up because you can't go to half of a game you can either go to 12 or 13. so if you go to 12 it's not worth it if you go to 13 it is okay that's why 20 is going to be c let's move on at an ice cream parlor kim can choose between two cones four flavors five toppings and three syrups if kim must choose one item from each category to make her ice cream sundae how many ways can she make her ice cream sundae of choice so it's asking for unique combinations the way you're going to do this is just multiply each of the options the number of options together and you end up getting i believe 120 21 should be e which of the following is the product of these two numbers so these are complex numbers you can find the product by just foiling and treating i like a variable and at the very end we'll apply the property of i squared is equal to negative 1 to get our final answer okay after we foil so first we're going to do 4i times 2i again that's just this then you do outer the two outer numbers plus 4i times negative 3. then the inner number is 5 times 2i and then the outer numbers 5 times negative 3. okay this will give you 8 i squared minus 12i plus 10i minus 15. here is where you can apply the property so remember i squared is equal to negative 1. and then these two numbers in the middle they both have an i term you can't really do anything with just an i term so you're gonna have to leave that uh there but you can combine the two terms okay treat it like as if it was x so this is going to turn into 8 times negative one minus two i minus fifteen eight times negative one is going to be uh negative eight so you end up getting negative 22 sorry negative 23 minus 2i so your answer for 22 is going to be d lines a and b lie in the standard coordinate plane if both lines are parallel and the equation for line a is this which of the following is the equation of b if it passes through the point two comma eight okay so imagine uh we know that they are parallel which means that they are going to have the same slope so the first thing we know about b already is it goes through and has the same slope the only other thing we need to identify is that y intercept okay how are we going to identify the y intercept we can plug in values for the y-coordinate and the x-coordinate and those values are right here and then what that'll do is that'll leave us with just one unknown and we can solve for that one unknown i'll just label it as a question mark okay so this is our y value or essentially we'll label as b because it's the b equation and this is our x value okay so 8 is equal to 3 times the x value 2 again minus question mark okay we subtract 6 on both sides sorry we're adding this so we get 2 equals question mark so your question mark is 2 your equation for b is going to be 3x plus 2. 23 is b scientific notation if it's multiplication or division that you're doing the best way to do this type of problem is to divide the problem into two different problems so we first do the just the decimal part the easy part you can do that on your calculator actually you can do the whole thing on your calculator uh if you just plug it in i'll show you how to do that but if you're not doing it on your calculator you can do 5.1 divided by 1.7 that's equal to 3. so you know you have to have a 3 in your answer that's the easy part i mean that's really not what the question's testing you on the second part if you use properties of exponents do it by hand this turns into since they have the same base of 10 you can just subtract the exponents so when you have 10 to the x divided by 10 to the y your answer is going to be 10 to the x minus y okay in this case x is equal to negative 9 y is equal to negative 12. so your answer this is going to be 10 to the negative 9 minus negative 12. don't get your signs mixed up just because there's extra negatives here doesn't mean that you need to you can get rid of that middle negative there that middle negative is the most crucial part of that whole problem of the whole process here okay because what's going to happen this is going to turn into a negative 9 plus 12. when you have a minus a negative it turns into a plus think of it as a plus okay so this becomes 10 to the third 3 times 10 to the third all right simple as that your answer should be this right here now the other way to solve this in seconds using your calculator is to plug in 5.1 capital e there's a capital e button on your calculator if depending on which calculator you have you can figure out how to press that negative 9 divided by 5.1 capital e or sorry we're doing 1.7 on the numerator denominator minus 12. okay this will give you three capital e minus three oh sorry plus three three capital e three and that just corresponds to three times ten to the third the capital e just is just short for ten to the really that's all it is it's the way your calculator quickly interprets that you could also type the whole thing out with the exponents if you want to type it all out just kind of make sure you do your parentheses properly so you don't mess things up the average of seven test scores is 85. if the lowest score is dropped the average of the remaining increases to 88 which of the following is the value of the lowest score that was dropped so if we remember the equation of an average is what average equals the sum of numbers divided by the number of numbers okay so i know that the original average is equal to 85 so i'll just write that here the original average is 85 original sum was 7 sorry the original number of numbers is seven uh so the sum has to be 85 times seven right in order to make this all true and make it all work it's 595 okay that's 85 times 7. all right now it's telling us that if we get rid of a certain test score the average i'm going to create a new average average 2 this goes up to 88 the sum we don't know what happens and the number of numbers is obviously 6. that means your sum does what your sum changes to 88 times 6 in order for this to be true that's 528. what i can do and if i want to find the value of that last test i can just subtract these two numbers so 595 if you do that on your calculator and you subtract 528 you end up getting 6 d7 and that's the value that's the difference between the 7 and the 6 test scores the difference was just that one square that was dropped okay another way to do that is to create put all of that into one equation so you could create one equation with these six tests and then this extra test that you don't know have the extra test be the unknown so the average will be 85. we're going to divide that by 7 right just write out your sum as 88 times 6 plus x where x is the unknown dropped so you multiply both sides by 7 you get 595 equals 88 times 6 plus x really what you're doing is 595 equals 528 remember these two numbers we just calculated them and then your final answer is just going to be the difference between these two numbers okay that's how you do this one a couple different ways to do it in a bag of marbles there are 22 red marbles 36 blue 38 green if greg wanted to increase the probability of drawing a red marble to one third how many red marbles would he have to add to the bag so what's the current probability of drawing a red well the probability of anything if you don't know how to calculate probability of an event is the probability is the number of desired outcomes divided by the number of total outcomes so the probability of drawing a red currently is the number of desired outcomes 22 divided by the number of total outcomes which is just the sum of all of those marbles how many marbles are there 36 let me just add that on my calculator plus 38 plus 22 96 and this is less than 1 over 3. i think this equals something along the lines of 0.27 so it's .22 0.229 okay so that's obviously less than 1 over 3. we need to increase now how do we increase the probability it says by adding red marbles to the bag okay so if i add red marbles to the bag what does that mean if i add one what happens my probability of red when we do plus one is what 22 plus one right because i added one red marble divide by 96 plus one because again in the numerator i added one red marble and in the denominator i also added a one red marble to the total right because that increase what does this give me gives me 23 divided by 97 which is 0.237 compared to the 0.229 that i was at before this is definitely closer which means that if i keep adding them i'm going to keep getting closer and closer to my desired value of 0.33 repeating right now you could just keep going like this do 22 plus 2 over 96 plus 2 or really just do check each of these answer options so like 22 plus 9 over 96 plus 9. you can go through and do that that might take you a minute or two but that's one way to solve it until you get the right value the best way to do it is to create an equation instead of checking values you check an unknown number x and you find what does that x value have to be for me to equal a probability of 1 over 3. so take that same equation 22 plus something divided by 96 plus something has to equal 1 over 3 right the probability has to equal that just make that something x what do i have here i have one equation with one unknown so i can solve for that unknown let's do some cross multiplying and dividing so i'm going to multiply both sides by essentially 96 plus x and then multiply them by 3 as well so cross multiply 3 times 22 plus x equals 96 plus x i can distribute the 3 out 66 plus 3x equals 96 plus x i can subtract 66 on both sides subtract x on both sides this turns into a 2x this turns into a that means i need to add 15 marbles to make the probability of a red one out of three and do you want to check the answer let's do it so probability of red is equal to 22 plus 15 over 96 plus 15 that gives me i believe 37 over whatever 37 times 3 is 111 and is 37 divided by 111 equal the desired value yes it does it equals 0.3 repeating okay because 37 times 3 is equal to 111 that's how you solve this if g f and h are positive integers such that f to the h is equal to y and g to the h equals z then what does y z equal so y z is going to just equal f times f to the h times g to the h okay simple as that but the thing is that's not one of the answer options so how do we combine this and get one of these well the thing is when two numbers have the same exponent for example two to the third times six to the third you can actually just multiply the two bases together and raise the entire product which is 2 times 6 in this case to that exponent okay that's a property of exponents so what you can do in this case is just multiply fg together raise it to h and that's what this is now you can't use this answer option for example because it might be tempting to add the two exponents together h plus h equals 2h but you can only do that when you have multiplication with the same base okay and the exponents might be different so for example if i had x to the third or if i had if the problem was uh let's say f to the h times f to the a then i could do f to the h plus a okay because they have the same base but when they have this different bases in this case f and g but the same exponent you can multiply the bases together okay so if these bases are are um the same you mult you combine the exponents if the exponents are the same you can combine the bases okay think of it that way scientific notation uh again you see it here and we're adding this time you can quite literally plug this into your calculator and it might if you have a graphing calculator it might just give you an answer in the form something something something e something something something um and if it does then you essentially know how to interpret that i did that in the problem up here how to interpret scientific notation on a calculator but assuming you're not doing it that way let's do it by hand just the proper addition with the way that they teach you in school let's convert this to scientific notation so in order to move the decimal from here all the way to 8.4 what do i have to do 8.4 times 10 to the one so i go 1 2 3 4 5 6 7 8 9 10 11. 6.3 times 10 to the what one so i'm going 3 6 9 10 11. okay cool so i moved my exponent um now what i'm doing is i'm adding these two together so i can just add since they have the same bases i can just add the exponents together like this so 8.4 plus 6.3 gives me 114.7 times 10 to the 11. that's not one of the answer options but it's close to one of these right it's definitely not this this is different um so between b and c how do we know which one to choose so we need to move the decimal whenever you write scientific notation it has to be the decimal the number in front of the decimal has to be in the ones place you can't have 14.7 it has to be 1.47 okay so how do i know if i change it to 1.47 what does that exponent on the base of 10 become well the way you're going to know is this is a very easy way to think about it if i take this number and i make this one smaller that means i have to make this one larger to compensate bigger okay so i made this smaller that means i'm going to make this bigger by by how much it's by the same factor that i made the other number smaller so what did i do i divided this by 10. that means i have to multiply this by 10. so if i divide by 10 here i'm going to multiply by 10 here okay meaning i take the original and i multiply by 10. this becomes 10 to the 11th times 10 to the first and that essentially just becomes 10 to the 12th 1.47 times 10 to the 12th very easy way to think about that you'll never get it wrong think of it as compensating what fraction lies exactly halfway between one half and two over three you can go ahead and plug in decimal equivalents of these so 0.5 plus like you can go on your calculator and plug in 1 divide by 2 plus 2 divide by 3 and then take that answer that it gives you and then divide it by 2 and you'll get a decimal the decimal will be whatever the equivalent of so it'll be 0.583 repeating and then what you could do is you could just check which of these equals 0.583 just off the top of your head you know that this won't this won't it might be one of these three okay very quick very easy way to back solve the problem now if you don't want to do it that way if you want to understand how to actually do it using the fractions and combining denominators and whatnot so in order to add you have to first add these together and then you have to find the average by dividing by the number of essentially fractions there are okay so the first step is adding it together in order to add it together you have to have the same denominator so a common denominator that you can use is just six just multiply the two denominators together i'm going to multiply this by 3 over 3 multiply this by 2 over 2. why do i do this why do i multiply by these seemingly useless fractions it's because i want the denominators to be the same so i'm going to multiply essentially by a fraction that has that this entire fraction is just equal to one same here it's just equal to one but it has a denominator that is the same as the other fraction i'm compensating essentially for the missing factor in the denominator so i have to think how do i go from 2 to 6 well i just multiply by 3. and since i don't want to change the value of the entire fraction i can just multiply the numerator by 3 as well what this gives me in this case is 3 over 6. it's the same number 1 over 2 3 over 6 it still equals half but i rewrote it with the denominator that i want so that i can add it to the other number with again that'll also have the denominator that i want okay in the other case you end up getting four over six seven over six is your answer and now again keep in mind you have to divide seven over six by two which is the same thing as seven over six times one over two seven over twelve if this is this then what does x equal well if i'm solving this i notice the numerators are the same so they essentially cancel out but the denominators are different i can make the denominators equal to one another divide both sides by square root of 13. x equals 13 over square root of 13. what does 13 over square root of 13 equal it's just square root of 13 because square root of 13 times square root of 13 is equal to 13. okay you can think of it that this as square root of 13 squared on the numerator all right otherwise do it with your calculator just to confirm um you can do this all in decimal form and then check which decimal answer option equivalent like convert all these to decimal form by just plugging them into your calculator if you do the problem in decimal form if you calculate this in decimal form you can find the right answer just by matching decimals james is a notebook with 100 pages he used eight pages of the notebook and lets his four sisters split the rest of the pages evenly amongst themselves how many pages do each of the sisters get to use so he uses eight pages so immediately 100 minus a like for example if we want to find how many pages are left right after he uses a pages if i use 23 pages for example and there's originally 100 then that means that the leftover is going to be 100 minus 23 pages which is 7 pages it's not going to be a minus 100 because that would give me 23 minus 100 and there's not a negative number of pages left which is going to be a positive number okay so we're going to subtract 100 minus a okay that's how many pages are left now he has to let the rest of his sisters split evenly those pages how many do each of them get well it's just going to be that number divided by 4. in the case of a equals 23 it'll just be 100 minus 23 which is 77 divided by 4 which is going to be some decimal they can split some pages here and there but the point is this is how you do it it's not divisible by 4 sorry it's not it's not divisible by 25 because while 100 divided by 4 is 25 that doesn't mean that we're dividing the pages into 25 parts there's no reason to divide into 25 parts we're dividing the uh 100 pages or whatever 50 75 pages into four parts given the following triangles that have equivalent corresponding angle measures which are the following does x equal if they have equivalent corresponding angle measures they are similar triangles which means that essentially one is a scale model a skill version of the other when one is a scale version of the other the ratio of side lengths from triangle to triangle are cause is constant okay so if i can find one ratio which in this case is times three that means that this other side length here since it's corresponds it's the left side length in this orientation is also going to be times three this is also going to be times three so three times three is nine four times three is twelve okay these two triangles hash out like this so your answer for x is going to be nine find the value m that makes the following two lines perpendicular so perpendicular lines have slopes that are negative reciprocals of one another so in this case if i have 3 really that's just equal to 3 over 1. so in order to get a negative reciprocal let's just take it step by step i first have to make this negative and then i have to take the reciprocal by flipping the numerator and denominator so if a line has negative one over three versus a slope of three those two lines will be perpendicular they can look like this they can look like whatever it will form a perfect x a squared x a cylindrical can of soup has height six inches and unknown radius given that the volume of a cylinder is known by this equation where r is the radius of the cylinder and h is the height which of the following is the radius of the can in inches if the following if the volume of soup in the can is this and the can is half filled okay so r is the unknown we're solving for that the volume they give us 27 pi over four the only thing is how do we get height they don't give us a height value i mean they give us this but this is the height of the can not uh the height of the fluid in the can because the can could be this high and the fluid could be just down here right so this is going to give us an inaccurate value if i use that value versus if i use this value for my height we're going to get a completely different value for h sorry for r the thing is it says the can is half filled so relative to the six inch total we are actually at three inches height of fluid in the can okay so i'm just going to make this equal to 3 and then solve for r so let's first solve this for r by dividing v so r is going to be equal to v over pi h square root if you just do some algebra you divide both sides by pi and h and then take the square root this is going to give you when i plug in the numbers with h equal to 3 and v equal to this number let's see what we end up getting 27 divided by 4 divided by 3 all to the half end up getting 1.5 which is right here okay so it's 1.5 inches uh in radius which makes sense for a can right as opposed to this very large number for a radius those two do not make sense a computer mouse's position tracking accuracy is directly related to its battery of power given the following table of values which are the following linear equations correctly models the mouse's position tracking accuracy as a function of battery accuracy a battery power okay so i have a table it might be linear i mean judging by the answer options it looks to be linear so how can i take this table and uh turn it into a function well if it's linear right let's let's first identify if it is linear so the way we're going to know if it's linear is if we notice a constant change in the x values which it is it's minus 50 minus 50. or sorry i guess that doesn't really tell us if it's linear we can just assume it is based on the answer options since it's not a constant change that doesn't really like what i was looking for is if it's a constant change here and a constant change here then you know it's linear um but instead what you could just do is you can find the slope between these two points and then compare it with the slope between these two points it will be equal because i've calculated it before um you can check if you'd like and it says they're linear so forget everything that i just said all right now how are we actually going to do use that information to solve so if i want to get a linear equation i need to have a slope and a y-intercept okay so the slope i can calculate just by doing m equals y 2 minus y 1 divided by x 2 minus x 1. if i identify this as my x and this is my y i can solve with just these two points for example so y two y one seventy minus ninety divided by x two x one fifty minus a hundred equals two over five okay so my slope has to be 2 over 5. this is the only answer option my y intercept really doesn't even matter okay it's a very simple problem the sum of all the edge lengths of a cube is 36 inches which of the following is the cube's volume so let's draw a cube really quick do this properly if i draw out a cube and i conceptualize this i can see all the edge links and know how many i have to add together okay so just pretend this is a cube it's not perfect um so i immediately already see one two three four one two three four five six seven eight nine uh but that's not all the edge lengths because there's also edge lengths on the back that i can't see so the cube goes down here this comes out here goes there as well okay so how many total edge lengths are there 1 2 3 4 5 6 7 8 9 10 11 12. okay and i think i've marked all of them another way to think of it is you have a square on top okay that's four sides you have a square on bottom that's another four and you have these four vertical things holding the two squares together again that's 12 okay so if the sum of the edge lengths is 36 that means that 12 times the length is going to be 36 that gives me an edge length of 3 for each okay now why did i just calculate all of that it's because the volume if i want the volume what's the volume of a cube it's the edge length cubed since i just calculated the edge length using the fact that there are 12 of them and that the sum of them is 36 i can just cube that value and get 27 inches cubed as my volume of the cube one of the following systems of inequality is depicts the region inside of a circle and above a line to be shaded which one is it so if you've watched the skill video for this skill you'll know that if it's shaded if the circle's shaded inside that means that when i write out the circle equation which is like this for example that the side of the equation with the x's and y's has to be less than the side of the equation with the radius squared again the radius squared here is 10 okay so in this case if i'm shading inside of a circle this side the x's and y's side of the equation has to be less than so that means c and d are out because they have greater than okay if it's greater than it would be outside the circle all right now above a line what does this mean so if i draw any line like this one here if we're shading above it how do we know whether we should shade above what does that sign look like is it going to be greater than or less than well intuitively it would be greater than but can you confirm right how would you confirm that mathematically one way to think of it as this equation is located here right now there is a y-axis here and the y-axis upwards in the upwards direction it goes on to positive infinity and in the negative direction it goes on to negative infinity okay when you draw your graph when you shade the side of the graph that has positive infinity if you're shading the side of the graph that has positive infinity in this case above that means that your shape that your sine is y is greater than okay when you write out your equation if it's a very long equation as long as the side that has y is greater than that means that you're going to shade on the side of the equation on the side of the line that has the positive infinity think of it as if y is greater than then y could literally be infinity y could be all the way up here here here you know it could go on forever it could also just be right above the line it could also be right here okay so again if the shot if the side of the line that has positive infinity is being shaded that means you're shading uh a greater than equation okay for more detail on this check out the skill video but essentially we need to have a greater than sign here that means that this is out we can't have an equal sign because then nothing would be shaded so that means your answer is a in the interval here sine theta equals cosine theta when theta equals theta equals what okay so sine theta and cosine theta when do they equal one another let's use the unit circle so i'm not actually going to draw the circle but i know that this is 0 pi over 2 pi 3 pi over 2 and then it comes back to 2 pi right so we go around i'll just draw the circle why not all right when theta equals what so i know just based on the answer options we're looking at multiples of pi over four so every 45 degrees basically we're looking at here here here here okay so would sine 45 like this is 45 degrees this is 135 degrees this is uh 225 degrees this is 315 degrees alternatively you can think of it as pi over 4 3 pi over 4 5 pi over 4 7 pi over 4. okay the easy way the brute force method to solve this is to just plug in sine of 45 sine of 135 sine of 225 sine of 315 cosine of 45 cosine of 135 cosine of 225 cosine of 315 and c make a table of all the values and see which one's equal and which ones don't you'll find that 45 aka pi over 4 and 5 pi over 4 are equal to one another so why aren't these two equal well if you remember the acronym a let me draw that again astc let me make it a bit more let me make a new graph really quick so what does astc mean if you draw your coordinate axes each of these corresponds to which of sine cosine or tangent is going to be positive and which chord in which quadrant so in quadrant one which is here in quadrant one all three of sine cosine and tangent are positive in quadrant two only sine is positive okay so if i do like for example any angle in quadrant two like um let's do 120 degrees sine of 120 is positive but cosine of 120 and tan of 120 are both negative in the tan quadrant aka quadrant three um only tan is positive sine and cosine are negative and here only cosine is positive so now why do i bring this up to you in quadrants like this and this sine and cosine have opposite signs in quadrants like this and this though they have the same sign so in order for them to be equal they have to have the same sign not just the same value at all four of these purple points they have the same value but not the same sign okay that's why you need to make sure that your sine equation has the same sign as the cosine and that's how you do this this particular problem use the figure below okay so the figure below depicts a control tower's area of jurisdiction which has a radius of 50 miles so this is 50 miles looks like we have a few planes so plane one plane two and plane three are three different aircraft at the very edge of the jurisdiction meaning they're all 50 miles away from the tower they are directly west north and east of the control tower respectively so we're going to use that to answer these two or three questions which of the following is the distance in miles between plane one and plane two okay so plane one and plane two is that's the distance we're looking for um i know this distance is 50 because again it told us that plane two all these planes are at the edge of the jurisdiction and this is directly north okay that's straight north this is straight west it says it's straight west right west north that means it's also 50. what i've just made here is an isosceles right triangle um i can solve for this last dimension using pythagorean theorem or just the uh my 45 45 90 triangle rules i'll know that this is 50 square root of 2 or i believe that's um yeah it's you can also find the the equivalent of that and a decimal form if you find the decimal form and you look at the answer options and you're not sure which one to choose just find the decimal equivalence of the of the answer options mainly just these two and see which one it matches it should match answer option b for question number 39. plane 1 is heading east at 300 miles an hour plane 3 is at a lower altitude heading east at 200 miles an hour after how many hours will they be directly above one another okay so plane one let's just kind of draw that out plane one is going this way i'm sorry east is the other way so east is this way at 300 miles an hour okay plane two is doing the same direction but it's at a lower mile per hour value so um when will they be directly above so that's kind of a hard question the way you can conceptualize this is if i made an equation a linear equation of the horizontal position of the aircraft where the slope is the velocity t is time and then the y-intercept is the initial x-position of the aircraft which would be relative to the control tower so let me say x on initial right i can create two equations one for each aircraft because they each have their own initial x value and their own velocities t and x will be my dependent variables essentially i've just made a y equals mx plus b except i made x a t because we're working with time that's my kind of variable that i'm inputting the independent variable um and then the the dependent variable is going to be x which is the again horizontal it's the east-west direction of these aircraft okay so let's go ahead and make an equation for plane one so for plane uh we're sorry yeah plane one so the x direction of 1 is going to be the velocity of plane 1 300 miles an hour times the number of hours plus the original horizontal distance so let me in order to find the original x value of each let's just say we have a coordinate plane here x y right so the original x position of plane 1 is negative 50 because this is 50 miles this way the original of plane 3 is positive 50 because it's 50 in that direction relative to the origin okay we don't really care about the y value so just actually just ignore that we're working with one dimension here okay so where i'm gonna for my x-intercept essentially my x i value i'm going to make this a negative 50. okay for the second plane for the it's really plane three so i'm going to call it x3 i have a 200 mile an hour velocity and i'm 50 miles in the right so i'm positive 50 away okay so what i can do is if i make these two equations equal to another one another i can find when x1 and is equal to x3 essentially at what t value do the x positions of these two planes align i can go ahead and make the equations equal add 50 on both sides minus 200 on both sides 100 t equals 100 so t is equal to one hour and that's your answer here okay after one hour they will be at the same position you can think of it as after one hour since these are currently 100 miles apart and this one's traveling at 300 this one's traveling at 200 after one hour this one will be 200 further in this direction and this one will be 300 further in this direction it will have traveled 300 miles aka 100 plus this 200 whereas this plane will have just traveled 200 so really plane 1 is compensating that 100 because of that extra 100 mile an hour that it's traveling at plane 2 is circling clockwise which is this way around the control tower's edge of jurisdiction until until it is directly east of the tower at whit which point it turns straight towards the control tower and makes a landing assuming the final distance from the tower is zero how far does plane 2 travel from its depicted position to its final position depicted position of plane two okay so it says plane two is traveling clockwise until it's directly east so east is here it travels clockwise until it's directly east and then it goes straight towards the tower and essentially makes this path so we have path one and then we have path two we need to find the distance of path one and path two and add them together okay now knowing that path one is uh we know that the radius is 50. so how can we get path one well we know that this entire thing is a circumference so if i take the one-fourth of the circumference i know what path one is as for path two that's just going to be the radius which is going to be 50. so that's the easy part so i can immediately start canceling out answer options i should have a 50 in my answer option so it's either 100 pi or 25 pi that'll be based on path 1 which is going to be the circumference right so circumference is what 2 pi r which is going to be 2 times pi times 50 which is going to be 100 pi now again don't make the mistake of making this your answer because 100 pi is one of the answer options but it's not correct because we're not traveling the whole circumference we're traveling one quarter of the circumference so we have to take one quarter of the circumference and that'll give us 25 pi and that's equal to path one so path 1 plus path 2 is equal to 50 plus 25 pi that's 41. which of the following is a rational number so let me first define a rational number a rational number is any number that can be written as a fraction of whole numbers so for example square root of 3 over 2. this is not a rational number because square root of 3 is not a whole number square root of 3 is a never ending decimal i is not a rational number because it's an imaginary number it's not a real concept it's a square root of a negative number however i squared is a rational number because it's negative 1 which is effectively the same thing as negative one over one it's a fraction of whole numbers they can be negative yes they can so okay just knowing that again numbers like pi e i these are all irrational because they're not whole numbers period it can't be written as a fraction of whole numbers okay but for example 5 over 7 it's a never ending decimal but that's a rational number okay so is this rational uh you can rewrite this as square root of 2 over 3 but the thing is that doesn't simplify to anything and that's a never never-ending decimal so because it's not a fraction of whole numbers it's irrational this is irrational because of pi this is irrational because this is whole but this is not sorry this is whole but this is not so that's wrong this is wrong because square root of 37 cannot be reduced to anything that's a whole number even though the denominator can and this is correct because this is just the same thing as 3 times negative 1. it's negative 3. it's a fraction of whole numbers effectively negative 3 over 1. okay that's how you do that problem given the following matrix which of the following values of t will make the determinant of the matrix negative 9 note that this is the determinant so how do we make the determinant of the matrix negative well let's just calculate what the determinant would be first so that's going to be t squared right a times d t squared minus b times c which is going to be b c so let me just walk you through that one more time effectively this is a this is b this is c this is d okay so a times d is t times t minus b times c six times t and we want to make that equal to negative 9. okay let me rewrite this t squared minus 6t equals negative 9. if you've seen a quadratic equation before then you might know where this is going i can move everything to one side and then i can factor this and solve for t you can also use quadratic formula i think factoring is a bit faster what are factors of 9 that equal negative 6 well the only factors that come to mind are 3 and 3 aka negative 3 negative 3 these two when you add them together they equal negative 6 when you add them so we're going to t minus 3 times t minus 3 is equal to 0. that means that t has to equal 3. if we want to get this criteria to be met we have to have t equals negative sorry t equals positive 3 43 is going to be d okay given that lines a and b are parallel and c and d are not parallel which they clearly do not look parallel which of the following is false so is angle one congruent to angle three these are congruent because of the opposite angles they form an x an extreme example is for example an x like this obviously these two are very small these are very large and you can use this as long as these two lines are straight which all lines are you can use this to conclude that the opposite angles are equivalent so these are equivalent five and seven like i said same thing if they form an x they're equivalent on opposites six and seven are not equivalent but the opposite angles across the x are okay six and eight same issue these two are equivalent because they're opposite of the x two plus four is it one eighty two plus four two so angle 2 is actually congruent to angle 4. okay the only way that b180 is if both of these were 90 degrees because they are the same however they're not 90 degrees or we don't know that they are necessarily because at least the way they're drawn it looks like this is a little bit more than 90. additionally what this is trying to say this is misconstruing the supplementary angles theorem which says that two angles that are on the same line but that are being split by another line essentially like 1 and 2 these form a 180 degree sum those are supplementary angles however 2 and 4 cannot be supplemented because they are opposite angles okay so it's something very different opposite angles are not the same they're actually very different from supplementary angles for that reason sorry d is going to be your answer 6 plus 7 on the other hand these two are essentially you see how they're adjacent they're touching and they are on the same line in that case they are going to be adjacent angles that are supplementary they add up to 180 degrees again supplementary is the same thing as adding up to 180 degrees okay when the two angles add up to 180 degrees a 12-foot ladder is propped up against a 10-foot high treehouse with the base of the ladder on the ground at the top of the ladder touching the treehouse's base as in the following figure which equation will give the minimum angle the ladder must make with the ground to reach the treehouse's base so we have two different angles here uh we're trying to look for an angle that's made from the ground to the treehouse's base so where's where's the ground this right because the ladder this is the ladder so let me just label that ladder ground where's the treehouse base is this the treehouse base not quite the treehouse space is an imaginary thing up here okay so we're looking for uh the minimum angle the ladder must make with the ground so the ladder is making an angle with the ground right here okay so whatever answer option we choose has to have a theta in it which means these angles of phi this is called fee those are all out okay so which one do we choose cosine or sine well if i do sine of theta that's creating a relationship between opposite and hypotenuse okay um that's going to be 10 over 12. so this is correct the ratio is correct cosine of theta is what it's it's adjacent so 2 squared of 11 over 12. now how do i choose between these two well the issue is sine is is not relating the ground to the ladder rather it's relating the height overall height the 10-foot dimension to the ladder and we're not asked about the height versus the ladder we're asked about ground and ladder which is essentially adjacent and hypotenuse since adjacent and hypotenuse adjacent and hypotenuse are what a cosine ratio describes we're going to choose the cosine ratio in this particular problem moving on a water tank has an initial volume of v equals zero gallons at eight am the volume of water in the tank is checked at various times as described in the table below and which time frame was the flow the highest so what you're going to do here is you're going to solve essentially for flow rate now what's flow rate it's the amount of flow divided by the amount of time so for example if 10 gallons flow in 20 minutes that means you have um or 10 gallons in 10 minutes that means you have one gallon per minute right 10 gallons in 20 minutes would be half a gallon per minute so in this case you want to find how many gallons per minute or gallons per hour are flowing um the way you're going to do that is you're going to look at each interval so you have an imaginary interval that's not plotted here in the in the table you have 8 am where you have 0 okay so you want to find in each interval what's the flow rate and find which one's the highest flow rate so if i look in this interval what's my flow rate i'm doing about 34 gallons in 2.4 hours okay so if you do 34 divide by 2.4 you get 14. 14. okay now in this interval i'm doing 14 gallons in about 68 minutes which is some other number of hours 68 divided by 60. so that's [Music] that gives me about 12.3 gallons per hour in this interval i'm doing one i'm doing a lot more gallons but i also have a much longer time frame about you know you could say seven hours let's just say it's 6.8 hours 6.7 12 and a half so again this one is still by far the largest interval from 8 am to 10 24 am and then my last interval at least the just between consecutive values 170 minus 133 gallons again i'm just comparing really what i'm calculating each time is 177 at least in this case right i'm calculating 170 minus 133 divided by the time that's elapsed you can think of it as minutes or you can think of it as hours okay we're really just doing the m m equals y 2 minus y 1 divided by x2 minus x1 you're doing that formula here okay so in that last interval how much time passes it looks like it's about three and a half hours so i'm gonna do 37 divided by three and a half that gives me about 10 point now the very last interval was uh it's listed here 8 am to 1 32 so it's going from here to here it's skipping the 1024 so we do 48 gallons in that amount of time over from 8 a.m to 11 32 that's three and a half hours so 48 divided by 3.5 48 divided by 3.5 that gives me 13.7 so by far the highest amount of water is in the 8 a.m to 10 24 am um interval okay so you kind of have to take time to check through these values but it's really a test of how you interpret flow rate and once you do that it's the rest of the calculation is pretty trivial the standard coordinate plane and its quadrants are shown below and which two quadrants does the line not appear in so what does that line look like it looks like this okay go straight through the origin um and it's gonna go straight through the origin like that it does not appear here or here so one and three you can also graph it on your graphing calculator if you're not sure and the quadrants are obviously labeled so that makes it a lot easier moving on below is the blueprint for a backyard patio design the center fountain is a square with side length 4 feet the entire patio is a square with side length 30 feet so 4 30. which of the following is the ratio of the fountains area to the rest of the patios area so we're comparing this area right here with this area right here okay so how do we find which one's bigger or not which one's bigger obviously we know that but we have to just find the the correct ratio the correct values okay so what's the patio area that's going to be um [Music] 30 squared but we're going to subtract the fountain area that gives us 884 okay it's just 900 minus 16. the fountain is going to be 4 squared and that's 16. okay so your ratio is going to be 16 to 884. so your answer for 48 should be b the sum of two numbers is 25. the smaller numbers squared so i'm going to let me let me first finish reading it then we'll write it out the smaller number squared is 14 less than half of 160. what's the larger numbers square okay two numbers so let's just create two numbers two variables x and y it says the sum is x is 25. the sum is when you add them together the smaller number let's just say is x so small you can assign it doesn't matter how you do it as long as you uh stick to what you assign it doesn't really matter how you assign them x or y okay the smaller numbers squared the smaller number is x squared is that's going to be equals 14 less than something so it's going to be something minus 14 half of 160. so 160 is 160 and half is just going to be half times that or essentially just 80 right so we the second equation gives us x squared is equal to 80 minus 14 and that's equal to 66 which really just tells us x equals square root of 66. i can plug that in into this equation and get y okay that's just going to give me one unknown so square root of 66 plus y is equal to 25 y is equal to 25 minus square root of 66 and that's going to be 25 minus seconds square 66 that's going to be 16.875 okay now if i the the question is not asking for the larger number it's asking for the larger number square which means what's y squared equal to y squared is going to be equal to 285 okay that's how you do this problem it's just a few terms you have to convert from algebraic from word to algebraic which of the following is the number of solutions the two functions below have in the region negative two is less than or equal to y is less than or equal to two so the way you're going to solve this is you're going to make the y's equal to one another or essentially make this equation equal to 2x minus 4 i can go ahead and subtract 2x and add 4 on both sides just so that i get everything on one side so that becomes x squared minus x minus 2 is equal to 0. now i can factor this x so i have to think about factors of negative 2 that gives me negative 1. so it's either negative 1 2 or negative 2 1. since i want a negative 1 i need to have the bigger number be negative so x minus two x plus one that means we have x equals two negative one okay now you might think all right these are both between the region here so is that correct no because these are x values so you're not going to just get you have to hold your horses it's not too real solutions uh at least not based on this information so let's save this piece of information let's finish the uh the rest of the problem but we're gonna we're gonna keep that in mind okay so how do we use that piece of information to actually get an answer for this what you're gonna do is you're gonna plug in these x values into my equation any equation for y you're gonna get the same answer for y no matter which one you do so might as well just use the simpler one so y1 we'll just call it is going to be 2 times 2 minus 1. i'm sorry minus 4. y2 is going to be 2 times negative 1 minus 4. so y1 equals whatever that is 0 y2 is equal to negative 2 minus 4 which is negative 6. now these are the y values that correspond to the x value so your coordinate pairs of your answers are 2 comma 0 and negative one comma negative six again zero negative six well now you can consider which ones fit in these regions this one does not so you have one real solution that fits in that region now you might be thinking um what about an imaginary solution an imaginary solution would exist if you had something along the lines of negative one plus or minus i okay where this uh or let me put it this way it would have to be negative it would have to be a plus or minus b times i where a is in this region a is a constant that is between negative two and two okay if that was the case then we would have uh one real and one imaginary solution but the thing is when you have one imaginary solution you usually have another so in this case you don't just have one you have just one real solution the other solution just doesn't fit in the range okay that's how you do this problem if the period of y equals cosine of 3 over 4x is 8 pi over 3 then the period of y equals cosecant 3 over 4x is what so you can go ahead and graph the equation you'll find that the equation has the same period of 2 pi over b for any and all trig functions where b is just the value in front of the x in your equation so if you were to write it out like this with a parenthesis the b value is just in front of the x okay so in this case b equals three over four so period equals two pi divided by three over four which is equal to 4 times 2 pi over 3 and that equals 8 pi over 3. same period same equation regardless of if you have second cosecant cotangent etc a regular tetrahedron is a 3d figure made of four equilateral triangles a certain regular tetragen has a surface area x and a side length l which of the following is the sum of its edge lengths expressed in terms of x okay that's a lot of information so we're essentially given surface area we are given and we want to find the edge lengths essentially expressed in terms of the variable x so how do we get anywhere near something like this the way we're going to do it is first of all let's try to draw a tetrahedron so we have a triangle it says equilateral so let me draw an equilateral triangle something like that i'm not going to draw it equilateral actually we're going to draw it like this so we have a triangle then we have another triangle and then we have a triangle in the back okay so you have one face here that's your first triangle another face here your second triangle another face on the bottom third triangle and then a face in the back fourth triangle okay so those triangles are all the same they're equilateral they all have the same dimensions okay that's an assumption that we're allowed to make um if it was if they all had if they were different triangles then we would not be able to call it a regular tetrahedron okay so the surface area is really just four times area of triangles all right and we can assume like it says here that they have an edge length of l so if i give a triangle here i say this how can i find the surface area of this triangle well i know triangle is just base times height and if it's equilateral triangle you don't know the height we're not given an h value but what you can use is the general form of area of a triangle where a and b are just any two lengths and c is the angle between those two lengths so if i say this is side a this is side b that means that this angle right here is angle c okay i know a and b are all going to be equal to l so these just substitute out as l sine c is what what is c so c in this particular triangle is the angle of 60 degrees because all the angles are 60 degrees it's an equilateral triangle right so area is going to be equal to half l squared sine of 60. what sine of 60 that's going to be equal to square root of 3 over 2. so area is going to be equal to l squared square root of 3 over 4. that's the area of one triangle now we have four triangles so the surface area total is going to be equal to l squared square root of 3. simple as that it's just l squared square root of 3. so surprisingly simple considering that we have four triangles we have different angles we have a sine ratio that's what we come out to okay and what does it say the surface area is equal to x okay now how do we take the surface area we want to look for the sum of the edge lengths so how do we take that and convert it to a sum of edge lengths the good thing is i have a drawing here so i can actually count the edge lengths in my drawing how many edges are there so i have one here one here one here i have three kind of going up to this one point right and then i have this triangle on the bottom one here one here one here in terms of side length so i have six edges right so the edge lengths are obviously they're just going to add up to 6 times l but we need to express 6l this is i mean this is technically your answer but we need to express 6l in terms of x so how can i do that i can express 6l in terms of x by solving this equation for l and then plugging it in here so solve x equals l squared square root of 3. solve it for l sorry so i'm going to divide it by square root of 3 and then take the square root so l so first divide by square root of 3. l squared equals x over and then i'm going to take the square root of both sides now i have a square root of a square root in the denominator here this effectively reduces to this right square root of the square root of three that can be reduced to fourth root of three because it's three to the half to the half which is just going to be if you have an exponent to an exponent you multiply the exponents that becomes three to the one fourth which is just the fourth root of three so that becomes square root of x divided by fourth root of three now you can see we kind of have a form that looks like this right and that's equal to l so this is your key equation right here and we're going to combine that with this so if this is what l is equal to what's 6l 6l is going to be equal to 6 times this thing so 6 times square root of x over 4th root of 3. square root of x over fourth root of three that's what that does okay pretty hard problem it's complicated uh but as long as you have the visualization down you're able to find surface area and you're able to interpret that there are six edge lengths the rest of it is just simple algebra moving things around definitely a hard problem that's why it's in the last 10 problems which of the following are the solutions for the below inequality so we have an inequality like this where there are essentially three components and you want to solve it for x what you have to do is isolate the x you can isolate the x by getting rid of the 18 and negative 2 and moving them to the other parts of the equation so let's start by doing that with the 18. subtract 18 from all three sides you end up getting negative 14 is less than negative two x is less than negative eight now the last step to isolate the x is to divide by negative two this is the tricky part so you can go ahead and divide by negative two but the thing is when you divide an inequality or multiply an inequality by a negative number the direction of the signs change they reverse so you're going to have nega 7 is greater than x is greater than 4. you can write this in a more conventional form by reversing it essentially moving the 4 here and flipping the sign again so 4 is less than x is less than 7. again these two are equivalent it's the same exact thing i just wrote it in in a reverse order so it's just easier to interpret smallest to largest right now this is your answer but none of the answer options are written in this form where it's combined you're gonna have to can take this inequality compound inequality and split it into two inequalities so we have four is less than x and then we have x is less than seven or essentially if i want to rewrite this this is x is greater than four x is greater than four x is less than seven that's what a is all right that's how you do this problem for the geometric sequence with first term 1 and third term 16 what's the sum of terms 5 4 5 and 8. okay the easiest way to do this problem you could do it with a of n equals a 1 r to the n minus 1 and you can find all the terms and do that summation that way probably the easier way is to just take the terms and write them out so i know that term one is is uh one term two is unknown term three is sixteen okay if you think what could this possibly be what could the common ratio possibly be let's just test some numbers so it can't be times two because if it was times two then this number would be four it can't be times three because if it was times three this number would be 27 or sorry it would be um it would be 9 not 27. what about times 4 can it be times 4 yes it can and it actually is so the common ratio r is going to be 4. and you can extrapolate that out to terms four five six seven and eight to essentially complete your sequence and very easily get all your values so we have 1 4 16 64 256 again i'm just multiplying by 4 each time 5 12 i think it's 10 24 how many do we have seven we need one more and then 40 96 okay so we need to add the sum of terms four five and eight so if you take a calculator you do um oh i just realized i made a silly mistake so this is actually not being multiplied by 4 each time i multiplied by 2 to get to 5 12. so 4 is going to give you 10 24 and then 4096 and then 16 uh i believe so what's 496 times 4 16 384 okay so i'm gonna add sorry these with these two four five and eight okay so 64 plus 256 plus 16384 is 16704 okay that's your final answer that's the sum of those terms what does this expression simplify to so i can use the foil method here so i'm going to multiply my first term sorry first terms in each parenthesis and then the outer terms and then enter and then the last so 6i times 6i plus 6 times 6i plus 6i times negative 6 minus sorry plus 6 times negative 6 okay this becomes 36 i squared this becomes right here 36 i we get plus negative 36i so i'm just going to write that as 36 i negative and then we have plus negative 36 that's minus 36. okay when we we can combine these terms uh what you'll notice is the middle two are just uh opposite sign same value so you can cancel those two out as for this i squared you remember that i squared is equal to negative 1. so multiplying by i squared is just the same thing as multiplying by negative 1. so your answer is going to be 36 times negative 1 minus 36 which is negative 36 minus 36 that's equal to negative 72. nope it is b which is 55. which of the following inequalities represents the values less than 8 units away from 3. so how do you solve this type of problem if you don't understand how to interpret this but you know how to solve absolute value inequalities you can go ahead and solve each one and based on the solution see which of these gives you a value or gives you a range of values that are all values less than eight units away from three or essentially values in the range eight plus three eight minus three um or sorry it should be you can think of it as three plus eight three minus eight which is going to be again eleven negative five okay so that's how you can do this type of problem and that's probably the best way to do it um the way to properly interpret it is the x minus something that part is just a locator so um the locator is 3 right that's the center point that we want to be based around so we won't have x minus 3 in the equation so that means that your answer is going to have to be d or e now as for the outside what does the outside number represent this represents the range or the distance that you can travel so we want to be from x minus 3 we want to be within 8 units of that number now how do we choose this sign well this just tells us do we want to be more than 8 units or less than 8 units like do we want to be do we want x minus 3 to be more than 8 units away or or do we want our x value essentially to be more than 8 units away from 3 or do we want to be less than 8 units away from 3. so if you choose greater than you're going to have again you're going to be outside that range we want to be within that range value is less than 8 units away so your answer should be e just based on this interpretation let's go ahead and check that see what we get when we plug these in so x minus 3 if i solve this is less than 8 x minus 3 is greater than negative 8. this becomes x is less than 11 and this becomes x is greater than plus 3 plus 3. negative 5. so between negative 5 and 11 are these all the values that are less than 8 units away from 3 yes they are what you'll find is if you solve any of these others you will not get values that match up because well this one has an equal to sign this says less than it doesn't say less than or equal to so for that reason this is wrong um as for a let me try that one so x minus 8 is less than 3. this gives you x is less than 11 which is correct but it'll also give you x minus 8 is greater than negative 3 which is going to give you x is greater than 5 which is clearly wrong 5 is not 8 units away from 3. so that's out you'll get a similar result here you'll get uh you'll probably get x minus five here x is less than five and that doesn't make sense you'll get and this is the same thing but it has the wrong sign so you'll get reversed signs here which again that does not make sense those are going to be values more than 8 units away from 3. given the following equations find how many z-intercepts f has as well as the number of solutions f and g have okay so z intercepts that f has based on this uh what are the z intercepts well i just set this equal to zero and the thing is it's already factored for me so i know that z has to equal three but the thing is that's the only uh z intercept that i was able to find so i only have one z-intercept the only thing we need to figure out now is how many solutions are there how are we going to figure out the number of solutions it's by setting f of z to be equal to g of z and finding what z values we get and how many we get okay so let's do that how many f how many z values so the way i can set these equal is uh i can expand this out this will become z squared minus six z plus nine and i set that equal to two z minus seven so z squared minus six z plus nine is equal to two z minus seven i can i wanna move everything to one side and then try to factor it that'll probably be the most logical thing to do add seven minus two z add seven so we get z squared minus eight z plus sixteen equal to zero i can go ahead and factor this fact what are the factors of four uh of 16 that equal negative eight i see negative four and negative four are a perfect fit so i end up getting z minus 4 z minus 4 is equal to 0 that means z has to equal 4. but the thing is i only have one solution again for this factored form of the quadratic which means that this is the only solution of the equation okay there's only one solution because there's only one factor in this comp combined solution here on a 10 question test a student has a one-quarter chance to guess any question correctly okay which of the following is the probability that the student guesses and score is zero um so if you have a one out of four chance of getting it right that means you have a three out of four chance of getting any given question wrong okay now none of the answer options here are equal to one out of four or three over four as a percent so why is that the case what you're gonna have to do here is not just find a fraction but you're going to take your fraction probability of 3 over 4 or 1 over 4 and raise it to a power that's because here you're dealing with essentially compound probability when there's a variety of events or independent events and you want to find the probability of all those events happening together in this case 10 questions being answered incorrectly together you have to multiply the probability by itself that number of times okay so you have your probability of getting a question wrong you need to raise that to the 10th power because this has to happen 10 times and whatever this value is that's your answer so take 0.75 raise it to the 10th and you end up getting about 5.6 percent so there's a if you take a ten question quiz and you're thinking worst case scenario if i what's the probability that i get everything wrong it's about one out of twenty which is uh strikingly high you know it's kind of alarming which is why you should study but that's how you do this type of problem if on the other hand they asked you if what's the probability of getting all the questions right it would be this and that's a much smaller probability so if you look at that 1 over 4 to the 10th that is 0.00009536 compare that to 0.056 this is like six magnitudes or five magnitudes smaller okay so it's much easier to get everything wrong than to get everything right the cross section of a paper towel roll is shown below so the inner radius is 0.75 inches the outer radius is three inches if the height of the paper towel roll is one foot what is the approximate volume of paper towel in the roll assume that paper towel sheets in the roll lay flat against each other okay so i mean you know what a paper towel roll looks like it's like this it's kind of vertical right so effectively that's what it is and we have a hole going through the middle and there's like a hole down there too yep that's what a paper towel rule looks like okay so um how do we it's effectively asking what's what's the volume of this region here how do you find the volume of that cylindrical region well what you can do is you take the vol the equation of the volume of a cube pi r squared times height and you modify the r squared value to be the difference in r squareds so r1 is going to be your bigger radius your maximum radius r2 is going to be your minimum radius okay um you can also think of it as you're finding the difference in area so the area of this large circle is going to be pi r i'm going to capital r for for large and then the area of the small circle is going to be pi r squared okay you can find the difference in those areas by just doing pi r squared minus r squared and then if you multiply that value by the height you'll get a volume of this region if it was extruded out vertically okay so if you just do that you'll get volume equals pi times uh three inches squared minus 0.75 inches squared uh multiply that by 12 feet um [Music] sorry 12 inches per foot sorry and you end up getting some value let's calculate that really quick so actually let's just do it in fraction form because i see the answers are still in fraction form so that'll be pi times 9 minus 3 over 4 squared times 12. that becomes pi times 12 times 9 minus 9 over 16 equals pi times 12 times we want to get a common denominator so i'm going to multiply 9 by 16. we get 144 over 16 minus 9 over 16. that becomes 12 pi times 144 minus 9 is 135 over 16. so if i multiply that 12 and 16 i end up getting 3 over 4. um [Music] 135 times 3 is what so we end up getting your final answer 405 pi over four okay again if this whole algebra thing at the end does not make sense to you don't worry about it just find the decimal equivalent plug this into your don't plug this into your calculator it's one step very simple find the decimal value and then just plug each of these into your calculator and match the decimal value with what you get here for which of the following sets is tan of theta less than or equal to one where theta is in radians so in order to solve this i'm going to draw out the unit circle it'll make it a lot easier if i visualize what's happening and i can see the problem as opposed to just thinking through it in my head okay so just to start we have the x-axis the y-axis i'm going to draw a circle actually we don't need the circle here i'm just going to plot the points so we have 0 degrees pi over 2 degrees negative pi so we have pi degrees and then we have 3 pi over 2 and then this goes up to 2 pi which is just the same thing as 0. okay now i want to find i want to analyze tangent at each of these like all around the circle and see what region or what are the various values how does tan like kind of vary throughout the throughout the unit circle okay the way i can do that is uh let's just analyze tan at each of these points okay so tan of zero is just going to be zero tan of pi over two is going to be what let's do tan of 90 see what i get i get positive infinity how do i know that well on your calculator that's not actually going to work what you need to do is you need to plug in tan of 89 for example and see what number it gives you tan of 89 is 57 it's just very large number if you do tan of 89.999 you end up getting 57 000 right so you're essentially looking at positive infinity here what about here tan of 180 it's zero and then tan of three pi over two again we would imagine it'd be negative infinity okay so we want to be less than or equal to one um i imagine these numbers are all negative right well actually hold on we know a s t c we know that tan's positive here so in that region where like tan is for example 225 what's tan of 225 we're at one so here tan is one if we look at 45 10 is also one if you again you can check that in your calculator what about here well here it says sign is positive remember astc all students take calculus this is the acronym that tells you in which regions of the unit circle which of the trigonometric functions sine cosine and tangent are positive so here in this quadrant one a all of the functions sine cosine and tangent are positive here only sine here only cosine okay so since tan is negative here it's it's we're not going to have we can include this region in in our analysis but you're not going to exclusively have this region so no that's not really super inclusive super conclusive right now let's just keep that in mind tan is going to be negative 1 here okay so that that definitely could be part of our answer same here 10 is going to be negative 1 at negative 45 degrees now again how do we get closer and closer to an answer um so between zero and one i know for sure this region works right because it i know tan is zero here and one here so this region for sure makes sense i can include this as part of my final answer what about this region right here because this is negative one i mean i guess it works right um so this part down here this if i do like for example i know this is 270 degrees right the 3 pi over 2 i can check if this region works in our in our final answer by just checking tan of 271 for example so that gives me negative 57. so this does work this is less than one okay this region works that means this region should work as well okay what about this region does that region work um again let's just check for example tan of 70. that gives me something like 2.7 so that doesn't work so i'm going to label this as red this is out of consideration because it's just too big what about here let's test something like tan of something greater than 90 tan of 110 can't be too much bigger because otherwise we'll end up in a different region so this does work it's less than one so that's fine um what about this region right here let's do something a little less than 180 degrees so tan of 179 yep it's negative as well this works too what about this region right here so if we're at zero going to one then this region would make sense right right because we're just going from zero to one that makes sense so anything here is going to be between 0 and 1. so it's called it's gonna be less than one but this region right here again since we know that this goes to uh i believe this goes to negative infinity so let's see what happens so let's see if i do tan of 269 57 so this actually becomes greater than the limiting value so this region does not work okay so we can't do that region right now let's take all that and bring it back to the answer options you don't have to do this first you can just solve this problem by checking each answer option but if you look at a a is effectively this region right here okay so it actually works if we look at b it's 0 to pi over 2. that region doesn't work because it includes this part from pi over 4 to pi over 2. this region here negative 3 pi over 4 is this up to pi over 4 which is that that does not work because it includes this red region here okay so that reason for that reason that's out negative one up to six so this is just again this is still radians uh it's a weird way of writing radians usually it's written as a as a multiple of pi but again negative one radian if you check what that is it's um so if you compare that with pi over four what's pi over four itself pi over four is just less than uh one radian so negative one radian's gonna be right here six radians though uh if we look at like pi right pi is here that's 3.14 radians so six radians is going to be literally we're going from this region all the way back to like here almost and that obviously includes a lot of red so we can't do that same thing happens here if we go if you go square root of 2 over 2 radians that's really just about the same thing as pi over 4 the value of it this is about equal to pi over 4. so this is saying if we go from pi over 4 about to here i think it definitely includes this region as well so for that reason that's wrong so we can't do this answer option either so the correct answer is going to be a because it includes only the yellow region that we can actually be a part of all right that's it for this video hopefully this was helpful

Channel: 5 Academy

Views: 87,887

Keywords: act math prep, act math, act math tips and strategies, act math review, act math formulas, act math tips, act math practice problems, act math prep 2020, how to improve act math score, how to learn act math skills, act math strategies, act math problem types, act math practice test, act math practice test solution, act practice exam, act practice test solution, act practice test breakdown, act math 2023, act math test 2023, act math test breakdown

Id: 7Bu1FEJB0P0

Length: 114min 32sec (6872 seconds)

Published: Tue May 24 2022