ACT Form 74F Math Answer Explanation


Choice C

$$\frac{x}{5}=\frac{6}{3} \\$$ $$ \Rightarrow x= 10$$

2017 April(74F) Math Question 2

Probability Practice

- One Event

2. The 35-member History Club is meeting to choose a student government representative. The members decide that the representative, who will be chosen at random, CANNOT be any of the 3 officers of the club. What is the probability that Hiroko, who is a member of the club but NOT an officer, will be chosen?

Choice K

$$p=\frac{1}{35-3}=\frac{1}{32}$$

2017 April(74F) Math Question 3

Logarithms Practice

- Logarithm Expressions

3. For what value of x is the equation 22′ 7 = 215 true?

Choice B

$$2x+7=15 \\$$ $$ \Rightarrow x= 4$$

Choice J

$$f(3)=5(3)^2-7(4\times 3+3)=-60$$

2017 April(74F) Math Question 5

Probability Practice

- One Event

5. A wallet containing 5 five-dollar bills, 7 ten-dollar bills, and 8 twenty-dollar bills is found and returned to its owner. The wallet's owner will reward the finder with 1 bill drawn randomly from the wallet. What is the probability that the bill drawn will be a twenty-dollar bill? DO YOUR FIGURING HERE.

Choice D

$$p = \frac{8}{5+7+8}=\frac{8}{20}=\frac{2}{5}$$

Choice H

$$40+2n=35+3n \\ $$ $$\Rightarrow n=5$$

2017 April(74F) Math Question 7

Angles Practice

- Other Shapes

7. In parallelogram ABCD below, AC is a diagonal, the measure of %ABC is 40°, and the measure of LACD is 570. What is the measure of LCAD ?

Choice G

$$\frac{8x-3}{x}=\frac{8\times \frac{1}{2}-3}{\frac{1}{2}}=2$$

Choice D

$$x=\frac{3+1}{2}=2 \\$$ $$ y=\frac{8+(-4)}{2}=2$$

Choice D

$$slope=\frac{-5-1}{2-(-2)}=-\frac{3}{2}$$

2017 April(74F) Math Question 12

Word Problems Practice

- Distance, Speed, Time

12. In Cherokee County, the fine for speeding is $17 for each mile per hour the driver is traveling over the posted speed limit. In Cherokee County, Kirk was fined $221 for speeding on a road with.a posted speed limit of 30 mph. Kirk was fined for traveling at what speed, in miles per hour?

Choice H

$$(x-30)\times 17 = 221 \\$$ $$ \Rightarrow x = 43$$

Choice B

$$x=\frac{12}{8}=\frac{3}{2} \\$$ $$ y=\frac{22-10}{2}=6 \\ $$ $$\Rightarrow x +y=7\frac{1}{2}$$

Choice H

Median = Average = 420 ÷ 5 = 84

→ sum of the 4 scores that are NOT the median = sum - median = 420 - 84 = 336

Choice D

$$||-8+4|-|3-9||=|4-6|=2$$

Choice K

$$x^{\frac{2}{3}}=\sqrt[3]{x^2}$$

Choice B

By rearranging the equation into slope intercept form, we have:

y =(4/7)x-(5/7)

Therefore the slope is 4/7.

Choice K

The sum of an odd integer and an even integer will always be an odd integer. For example, 1 + 2 = 3

Choice B

$$AB=\sqrt{32^2+24^2}=40 \\$$ $$\Rightarrow \frac{AB}{2}=20$$

Choice K

It is not given that the triangle is a right triangle, so you can’t use the Pythagorean theorem. The length of DF ranges from, not inclusively, (√30-3) to (√30+3).

Choice B

$$Area = (8\times 10 + 8\times15)\times2-60=340$$

Choice F

$$[w+(w+5)]\times 2=40 \\$$ $$ \Rightarrow w=7.5$$

Choice C

$$8\% \cdot60=\frac{1}{5}x \\$$ $$ \Rightarrow x= 24$$

Choice J

$$14x\geq 175 \\$$ $$ \Rightarrow x \geq 12.5 \\ $$ $$\Rightarrow x_{min}=13$$

Choice A

$$\frac{4.8\times10^{-7}}{1.6\times10^{-11}}=3.0\times 10^{-7+11}=3.0\times 10^{4}$$

2017 April(74F) Math Question 26

Angles Practice

- Circles

26. A circle in the standard (x,y) coordinate plane has center C(-1,2) and passes through A(2,6). Line segment AB is a diameter of this circle. What are the coordinates of point B ?

Choice H

74f

Vector CA which points from C to A has components given as <3, 4>. Thus, the vector that points from C to B = -CA = <-3, -4>. Hence, the coordinates of point B is (-4,-2).

2017 April(74F) Math Question 27

Quadratic Equations Practice

- Factoring

27. Which of the following expressions is a factor of x3 - 64 ?

Choice A

$$x^3-64=(x-4)(x^2+4x+16)$$

Choice H

$$\frac{4\times90-80+96}{4}= 94$$

Choice E

$$a^2=(-2.5)^2=6.25$$

Choice J

There is 7/9 of the original pizza remaining. Thus, each brother will get (1/3)(7/9) = 7/27 of the original pizza.

Choice E

$$30030=30\times1001=(2\times3\times5)\times(7\times11\times13)$$

2017 April(74F) Math Question 32

Perimeter, Area, Volume Practice

- Mixed Shapes

32. What is the area, in square inches, of the scale drawing of the park?

Choice G

74f

Area of the scale drawing = (28 + 40) · 16 / 2 = 544 square inches

Choice E

$$Length \ of green \ side=\sqrt{16^2+12^2}=20 \\$$ $$ \Rightarrow Perimeter = 20+28+16+40=104 \ inches = 104\times 1.5 \ feet = 156 \ feet$$

Choice H

$$\frac{40}{28}=142\frac{6}{7}\%$$

Choice C

The area of the walkway can be arrived at by deducting the area of bigger rectangular (30’ x 36’) from that of the smaller rectangular (30’ x 24’):

Area of walkway = 30’ x 36’ - 30’ x 24’ = 360 square feet

2017 April(74F) Math Question 36

Word Problems Practice

- 1 Variable Equations

36. Mrs. Smith will install a ceiling fan in each room of the cabin and will place curtains over the 4 windows. Each of the ceiling fang has a price of $52.00. The price of curtains for each small window (S) is $39.50, and the price of curtains for the large window (L) is twice that for the small window. Based on this information, which of the following values is closest to the total price Mrs. Smith will pay for curtains and ceiling fans?

Choice J

There are 3 rooms so 3 fans will cost 3($52) = $156. There are 3 small and 1 large windows so the total cost of the curtains will be 3($39.50)+1($79) = $197.50. Therefore, the total cost is $156+$197.50 = $353.50.

2017 April(74F) Math Question 37

Probability Practice

- More than One Event

37. Mr. and Mrs. Smith plan to roof the cabin on 2 consecutive days. Assuming that the chance of rain is independent of the day, what is the probability that it will rain both days ?

Choice A

Because the chance of raining on a particular day is independent of the chance of rain on another day, the probability that it rains two consecutive days is (0.2)(0.2) =0.04

Choice K

An irrational number cannot be written as a ratio of two integers. (F) evaluates to √(1/4) = 1/2. (G) evaluates to √(4) = 2 = 2/1. (H) evaluates to 2 = 2/1. (J) evaluates to √(16) = 4 = 4/1. (K) cannot be simplified further and cannot be written as a ratio of two integers.

Choice D

$$\tan \theta = \frac{4}{10}=\frac{2}{5}$$

Choice K

$$|2x-8|=2 \\$$ $$ \Rightarrow 2x-8=2 \ or \ 2x-8=-2 \\$$ $$ \Rightarrow x=5 \ or\ x=3$$

Choice A

There are 12 students who have scores between 65-70 but 13 who have scores in that range plus the range of 71-80 (totalling a range of 65-80). Thus, there are 13-12 = 1 student with scores between 71-80.

2017 April(74F) Math Question 42

Logarithms Practice

- Logarithm Expressions

42. The number of decibels, d, produced by an audio source can be modeled by the equation d = 10 log(7), where I is the sound intensity of the audio source and K is a constant. How many decibels are produced by an audio source whose sound intensity is 1,000 times the value of K ?

Choice G

For this sound, I = 1000K, so d = 10log(1000K/K) = 10(3) = 30

2017 April(74F) Math Question 43

Statistics Practice

- Mean, Median

43. Mario plays basketball on a town league team. The table below gives Nlario’s scoring statistics for last season. flow many points did Mario score playing basketball last season!

Choice C

$$score = 1\times 80 \times 75 \% + 2 \times60 \times90 \%+3\times60 \times 25\%=213$$

Choice F

The graph is moved to the right for n units whenever x is substitute for x - n.

2017 April(74F) Math Question 45

Perimeter, Area, Volume Practice

- Mixed Shapes

45. Toby wants to find the volume of a solid toy soldier. He fills a rectangular container 8 cm long, 6 cm wide, and 10 cm high with water to a depth of 4 cm. Toby totally submerges the toy soldier in the water. The height of the water with the submerged toy soldier is 6.6 cm. Which of the following is closest to the volume, in cubic centimeters, of the toy soldier?

Choice A

The volume of the toy soldier can be found by deducting the volume of the rectangular container after the submerging of the toy soldier from that before:

Volume of toy soldier = 8 x 6 x 6.6 – 8 x 6 x 4 = 125

2017 April(74F) Math Question 46

Perimeter, Area, Volume Practice

- Mixed Shapes

46. A box in the shape of a cube has an interior side length of 18 inches and is used to ship a right circular cylinder with a radius of 6 inches and a height of 12 inches. The interior of the box not occupied by the cylinder is filled with packing material. Which of the following numerical expressions gives the number of cubic inches of the box filled with packing material?

Choice J

Volume of the box = 18³

Volume of the cube = π(6)(12)²

--> Volume of packing material = 18³ - π(6)(12)²

Choice B

1 yard = 3 feet

1 feet = 12 inches

Area of the floor = 15 feet x 21 feet = 5 yard x 7 yard = 35 square yards

2017 April(74F) Math Question 49

Perimeter, Area, Volume Practice

- Mixed Shapes

49. The graph of a function y = f(x) consists of 3 line segments. The graph and the coordinates of the endpoints of the 3 line segments are shown in the standard (x,y) coordinate plane below. What is the area, in square coordinate units, of the region bounded by the graph of y .f(x), the positive y-axis, and the positive x-axis?

Choice B

The area is simply the area of the trapezoid spanned from x = 0 to 2 plus the area of the rectangle from x = 2 to 3 plus the area of the triangle from x = 3 to 5. Thus, the area = .5(4+3)(2)+(1)(3)+.5(3)(2) = 13

Choice J

x + y = 151

x=19+√y

--> x²-37x+210=0

--> (x-7)(x-30)=0

--> x = 7 or 30

If x = 7, √y<0, therefore x cannot be 7

--> x = 30, y = 121

-->y-x = 91

Choice C

The list of number is ordered from the highest to the lowest.

The median of the list is 25, therefore (30+X) / 2 = 25 –> X = 20

The mode of the list is 15, therefore Y = 15

The mean of the list = (41+35+30+20+15+15) / 6 = 26

Choice F

Substituting y for x², we have

s·x² + r·x – t = 0

For the equation to have two different solutions,

Δ = r² + 4st › 0

Choice A

$$a_{3}=13 \\ a_{4}=18 \\$$ $$ \Rightarrow d=a_{4}-a_{3}=5 \\$$ $$ \Rightarrow a_{50}= a_{4} +(50-4)\times d =248$$

Choice H

The Pythagorean trigonometric identity is expressed as:

sin² x + cos² x = 1

Therefore, y = sin² x + cos² x = 1

2017 April(74F) Math Question 55

Trigonometry Practice

- Graphing Trig Functions

55. What is the period of the function f(x) = esc(4x) ?

Choice E

The regular period of csc(x) is 2π. Thus, csc(4x) represents a compression of csc(x) towards the y axis by a factor of 4. Thus, the period will be 2π/4 = π/2

2017 April(74F) Math Question 56

Probability Practice

- One Event

56. At the school carnival, Mike will play a game in which he will toss a penny, a nickel, and dime at the same time. He will be awarded 3 points for each coin that lands with heads faceup. Let the random variable x represent the total number of points awarded on any toss of the coins. What is the expected value of x

Choice H

For the toss of a penny, the probability of it landing with its head faceup is 50%. If it lands with its head up, the awarded value is 3 points. Otherwise the point is 0. Therefore the expected value of the point awarded for the toss of a penny is 3 x 50% + 0 x 50% = 3/2.

The same can also be said of the toss of a nickel and a dime. Therefore the expected value of the total points awarded is 3/2 +3/2 +3/2 = 9/2.

2017 April(74F) Math Question 57

Matrices Practice

- Determinants

57. For what positive real value of k, if any, is the determinant of the matrix [ k 3 k 4

Choice B

We wish to find k for which k2-12 = k. Rearranging yields k2-k-12 = (k-4)(k+3) = 0. Thus, k = 4

Choice F

If i² = -1, then i^4 = (-1)² = 1. And notice that (i^4)k = i^4k = 1k = 1, k ∈ ℤ. In other words, i^4 = i^8 = i^12 = … = 1. Thus, we may conclude that 4|n (4 divides n; you do not need to know this notation).

2017 April(74F) Math Question 59

Trigonometry Practice

- Graphing Trig Functions

59. For 5 0 5 1, ‘sin 01 1 is true for all and only the values of B in which of the following sets?

Choice A

Recall that |sinθ| ≤ 1 for all θ ∈ R. Thus, we need only to find θ where sinθ = ±1. Going back to your unit circle, those values are θ = ±π/2

2017 April(74F) Math Question 60

Angles Practice

- Vertex Angles & Bisectors

60. Ray PK bisects LLPM, the measure of ZLPM is I lx°, and the measure of LLPK is (4x + 18)°. What is the measure of ZKPM ?

Choice K

74f

∠LPK · 2 =∠LPM

->(4x + 18°) · 2 = 11x

-> x = 12°

-> ∠KPM = 4· 12° + 18°=66°