ACT Form 70C Math Answer Explanation



2012 June(70C) Math Question 1

Statistics

- Mean, Median

1. The table below gives the heart rates, in beats per minute(bpm), for 8 participants in a fitness study after each participant completed 45 minutes of aerobic exercise. What is the mean heart rate of the
8 participants, to the nearest 0.1 bpm?

Choice C

$$\frac{1260}{8}=157.5$$


Choice H

$$4\times L=144$$ $$L=36$$


2012 June(70C) Math Question 3

Word Problems

- Cost

3. The price of a coat decreased from $50 to $40. The price decreased by what percent?

Choice D

$$\frac{50-40}{50}\times100=\frac{10}{50}\times100=20$$


Choice J

$$\frac{8}{50}=\frac{x}{500}$$ $$\Rightarrow x=80$$


Choice C

$$(8+12)2=20(2)=40$$


2012 June(70C) Math Question 6

Word Problems

- Fixed Cost + Variable Cost

6. At Nikki's Necklaces, the total production cost to roduce necklaces consists of an operational cost of
300 per day, plus a material cost of $10 per necklace produced. For a day in which n necklaces are produced, which of the following expressions gives that day's total production cost, in dollars, to produce necklaces?

Choice H

$$300+$10\times n$$


2012 June(70C) Math Question 7

Probability

- Combinations

7. The first question on a 2-question quiz offers
2 answers, and exactly I answer must be chosen. The second question offers 5 answers, and exactly I answer must be chosen. The quiz has how many possible combinations of answers?

Choice B

$$2\times5=10$$


Choice J

$$\Rightarrow 2x-12+x=36$$ $$\Rightarrow 3x-12=36$$ $$\Rightarrow 3x=48$$ $$\Rightarrow x=16$$


Choice D

$$3(4+a)=2(9+a)$$ $$12+3a=18+2a$$ $$a=6$$


2012 June(70C) Math Question 10

Sequences and Patterns

- Geometric Sequences

10. The 1st term in the geometric sequence below is-6 If it can be determined. what is the 6th term?
6, 12, -24, 48, -96, …

Choice F

$$12\div(-6)=-2$$ $$(-96)\times(-2)=192$$


Choice D

$$(-4,-9)\rightarrow(-4+4,-9+9)=(0,0)$$


Choice F

$$510\div(5\times12)=510\div60=8.5$$


Choice C

$$\frac{150}{0.75}=112.5$$


2012 June(70C) Math Question 14

Word Problems

- 1 Variable Equations

14. If cantaloupes sell at S1. 49 each or 3 for S3.90, how much is saved, to the nearest cent, on each cantaloupe by buying them 3 at a time?

Choice G

$$3.9\div3=1.3$$ $$1.49-1.3=0.19$$


Choice E

$$=(4\times3)(a^{1+4})(b^{2+3})$$ $$=12a^{5}b^{5}$$


2012 June(70C) Math Question 16

Equation of a Line

- Distance & Midpoint

16. In the standard(x,y) coordinate plane, the coordinates of the endpoints of DM are(11, 3)and(17, 15). What is the y-coordinate of the midpoint of DM?

Choice G

$$\frac{3+15}{2}=9$$


Choice E

She stopped.


2012 June(70C) Math Question 18

Angles

- Parallel Lines

18. In the diagram below, lines m and n are cut by transversal t.
Lines m and n are NOT parallel. which of the following statements must be true?

Choice G

$$\angle 1\cong\angle3$$ $$\angle 2\cong\angle4$$ $$\angle 8\cong\angle6$$ $$\angle 5\cong\angle7$$


Choice C

$$=5-6x+3$$ $$=8-6x$$


Choice J

An obtuse angle is a type of angle that is always larger than 90° but less than 180°.


Choice C

$$f(-5)=\frac{-3[(-5)^{2}+3(-5)+2]}{15(-5)+15}$$ $$=\frac{-3(25-15+2)}{-75+15}$$ $$=\frac{-3(12)}{-60}$$ $$=\frac{36}{60}$$ $$=\frac{3}{5}$$


Choice G

$$y=mx+b$$ $$2y=-3x+6$$ $$y=\frac{-3}{2}x+6$$ $$m=\frac{-3}{2}$$


Choice A

$$\sqrt{20}-\sqrt{45}$$ $$=\sqrt{5}\sqrt{4}-\sqrt{5}\sqrt{9}$$ $$=\sqrt{5}(\sqrt{4}-\sqrt{9})$$ $$=\sqrt{5}(2-3)$$ $$=-\sqrt{5}$$


2012 June(70C) Math Question 24

Word Problems

- % Interest

24. A formula for simple interest is I= Prt,  where I is the interest in dollars, P is the principal in dollars, r is the annual interest rate expressed as a decimal, and t is the time in years the money is invested. Which of the following expressions gives t when the annual interest rate is 5%?

Choice F

$$l=Prt$$ $$t=\frac{l}{Pr}=\frac{l}{P\times0.05}$$


Choice A

$$f(2005)=2f(1998)+100$$


2012 June(70C) Math Question 26

Trigonometry

- Pythagorean Theorem & sin cos tan

Use the following information to answer questions 26-28.
Shown below is right triangle AABC with the given dimensions in meters.
26. What is the length, in meters, of BC?

Choice K

$$4^{2}+6^{2}=x^{2}$$ $$16+36=52=x^{2}$$ $$x=\sqrt{52}$$


2012 June(70C) Math Question 27

Angles

- Collinear Points & Triangles

27. Which of the following statements about the measures of the interior angles in △ABC is true?
(Note: m∠ A represents the degree measure of ∠A)

Choice C

$$\angle A+\angle B+\angle C=180°$$ $$\angle …


2012 June(70C) Math Question 28

Trigonometry

- Pythagorean Theorem & sin cos tan

28. Which of the following expressions represents cos 8?

Choice G

$$\cos B=\frac{\overline{AB}}{\overline{BC}}$$


Choice B

$$g(1)=1^{2}+1=1+1=2$$ $$f(g(1))=f(2)=2+3=5$$


Choice K

$$=3^{3}\times x^{2\times3}=27x^{6}$$


2012 June(70C) Math Question 31

Trigonometry

- Pythagorean Theorem & sin cos tan

31. Marsha is making a sail for a model boat. In the figure below, the sail is represented by △ABC Base BC is 18 inches long, and legs AB and AC are each 15 inches long. The mast, represented by AD, is perpendicular to the base and extends 4 inches below the bottom of the sail. How many inches long is the mast?

Choice B

$$15^{2}-9^{2}=\overline{AE}^{2}$$ $$225-81=144=12^{2}$$ $$\overline{AE}=12$$ $$12+4=16$$


2012 June(70C) Math Question 32

Number Types & Properties

- Order of Values

32. Which of the following inequalities orders the numbers below from least to greatest?
3(3/7), -(-3.5),  3(2/5)

Choice K

$$3\frac{3}{7}=3\frac{30}{70}$$ $$-(-3.5)=3.5=3\frac{35}{70}$$ $$3\frac{2}{5}=3\frac{28}{70}$$ $$\Rightarrow 3\frac{2}{5}\lt3\frac{3}{7}\lt-(-3.5)$$


2012 June(70C) Math Question 33

Word Problems

- 2 Variable Addition

Use the following information to answer questions 33-35.
The table below gives the prices for dog grooming at Pretty Pooches.
33. Hoakoa brought 10 dogs to Pretty Pooches and paid
$260 to have all 10 shampooed. How many large dogs did Hoakoa have shampooed?

Choice D

$$S:x$$ $$L:10-x$$ $$x(20)+(10-x)35=260$$ $$20x+350-35x=260$$ $$350-260=15x$$ …


2012 June(70C) Math Question 34

Word Problems

- Cost

34. Jim took his large dog to Pretty Pooches for a haircut
3 times. Jim paid full price for the first 2 haircuts.
and had a coupon for the 3rd haircut, giving him a
15% discount on the 3rd haircut. How much did Jim pay for the 3 ha haircuts?

Choice H

$$55+55+55(0.85)=137.5$$


2012 June(70C) Math Question 35

Word Problems

- Fixed Cost + Variable Cost

35. When the owner of Pretty Pooches increases the price to have a small dog shampooed, the number of small dogs shampooed per day decreases. The expression ax+b represents the number of small dogs shampooed in I day whenever the price is x dollars per dog. The number of small dogs shampooed per day was 12 when the price in the table was in effect. The number of small dogs shampooed per day decreases by 2 for every $5 increase in price. What are the values of a and b?

Choice B

$$12=20a+b---(1)$$ $$(20+5)(12-2)$$ $$\Rightarrow 10=25a+b---(2)$$ $$(2)-(1)$$ …


Choice K

$$\frac{1}{1+\frac{1}{1+\frac{1}{2}}}$$ $$=\frac{1}{1+\frac{1}{\frac{3}{2}}}$$ $$=\frac{1}{1+\frac{2}{3}}$$ $$=\frac{1}{\frac{3}{5}}$$ $$=\frac{5}{3}$$


Choice D

$$y=0=-2x+8$$ $$x=4$$


2012 June(70C) Math Question 38

Quadratic Equations

- Factoring

38. A toy rocket is launched from the ground. Its height h feet above the ground, t seconds after it is launched is given by h=-16t2+48t. During the rocket's ascent at what value of t is the rocket 32 feet above the ground?

Choice H

$$h=-16t^{2}+48t$$ $$32=-16t^{2}+48t$$ $$\Rightarrow -16t^{2}+48t-32=0$$ $$\Rightarrow …


2012 June(70C) Math Question 39

Inequalities

- Number Line

39. Which of the following graphs illustrates the solution set for the system of inequalities 3x-2>9 and
-2x>-10?

Choice D

$$3x-2\gt9$$ $$\Rightarrow 3x\gt11$$ $$\Rightarrow x\gt\frac{11}{3}$$ …


2012 June(70C) Math Question 40

Equation of a Line

- Distance & Midpoint

40. A cab picks up a passenger at a point that is designated on the map as (7, -3) and is 7 miles east and 3 miles south of the cab company's headquarters From(7,-3), the cab travels 4 miles due north, 5 miles due east, and then 2 miles due west, where the passenger is dropped off. Which of the following is closest to the straight-
line distance, in miles. the cab is from the cab company's headquarters?

Choice J

$$(7,-3)\Rightarrow(7-7,-3+3)=(0,0)=headquarters$$ $$(7,-3)\Rightarrow(7+5-2,-3+4)=(10,1)$$ $$(0,0)\rightarrow(10,1)$$ $$\sqrt{(10-1)^{2}+(1-0)^{2}}=\sqrt{81+1}=\sqrt{82}\approx 9.05$$


2012 June(70C) Math Question 41

Trigonometry

- Pythagorean Theorem & sin cos tan

41. During a competition, a swimmer will be swimming the width of a river, from X to y in the figure below.
The swimmer wants to know how wide the river is The right triangle shows the measurements the swimmer obtained by walking 100 feet from X to Z, sighting to Y from both points. From this sighting. it is determined that the measure of /z is 40. How wide in feet, is the river?

Choice F

$$\tan40°=\frac{\overline{XY}}{100}$$ $$\overline{XY}=100\tan 40°$$


2012 June(70C) Math Question 42

Trigonometry

- Pythagorean Theorem & sin cos tan

42. The figure below shows a support wire for a television transmission tower. The wire, which is fastened to the tower at a point 125 feet above the level ground, has an angle of elevation of 65. Which of the following expressions gives the length, in feet. of the wire?

Choice F

$$\sin 56°=\frac{125}{?}$$ $$?=\frac{125}{\sin 56°}$$


2012 June(70C) Math Question 43

Perimeter, Area, Volume

- Mixed Shapes

43. A circle is inside a rectangle 6 inches wide and
8 inches long and is tangent to 3 sides of the rectangle.
What is the area, in square inches, of the circle?

Choice C

70c

$$\pi r^{2}=\pi\times3^{2}=9\pi$$


2012 June(70C) Math Question 44

Conics

- Circles

44. Which of the following equations represents the circle with center (2, 1) shown in the standard (x,y)
coordinate plane below?

Choice G

70c

$$r=1-(-1)=2$$


2012 June(70C) Math Question 45

Inequalities

- Graphing Inequalities

45. One of the following inequalities, where both constants a and b are positive real numbers, is graphed in the standard (x,y) coordinate plane below. Which inequality is it?

Choice B

$$m=-\frac{a-0}{b-0}=-\frac{a}{b}$$ $$\Rightarrow y\leq-\frac{a}{b}x+a$$


Choice F

$$r_{B}=\frac{2x}{2}=x$$ $$x:x=1:1$$


2012 June(70C) Math Question 47

Perimeter, Area, Volume

- Mixed Shapes

47 In trapezoid KLMN shown below, ∠M measures 45
and ∠K and ∠ N are both right angles. The given side lengths are in feet. Which of the following values is closest to the area. in square feet, of KLMM?

Choice E

$$1:1:\sqrt{2}$$ $$\Rightarrow 5\sqrt{2}:5\sqrt{2}:10$$ $$18\times5\sqrt{2}+5\sqrt{2}\times5\sqrt{2}\times\frac{1}{2}$$ $$=90\sqrt{2}+25$$ …


2012 June(70C) Math Question 48

Sequences and Patterns

- General Sequences

48. For the 2 o'clock hour on Milena's grandfather clock
(shown below), the bell is struck 2 times. For every 3 o'clock hour. the bell is struck 3 times. For every hour, 2 seconds elapse between consecutive strikes of the bell. For the 9 o'clock hour, how many seconds elapse between the 1st strike of the bell and the
9th strike of the bell?

Choice J

There are 8 pauses between …


2012 June(70C) Math Question 49

Inequalities

- Inequality Expressions

49. For positive real numbers a, b, and c such that 2a =3b and 1/b=1/2c, which of the following inequalities is true?

Choice A

$$2a=3b$$ $$\frac{1}{4}b\times4=\frac{1}{2}c\times4$$ $$\Rightarrow b=2c$$ $$2a=3b=6c$$ …


2012 June(70C) Math Question 50

Sequences and Patterns

- Arithmetic Sequences

50. This month Malcolm has $100 saved, and his goal is to have a total of $310 saved 12 months from now After adding to his savings next month. each month he will add SI more than what he added the previous month. For the next 12 months. Malcolm will not remove any money from what he has already saved What is the minimum amount of money Malcolm must add to his savings next month so that he reaches his goal?

Choice F

Because the amount he is …


2012 June(70C) Math Question 51

Number Types & Properties

- Complex Number

51. For all pairs of nonzero real numbers a and b, the product of the complex number a+bi and which of the following complex numbers is a real number?

Choice C

$$\because i^{2}= -1$$

The correct …


Choice G

70c

$$\frac{100}{4}=2r$$ $$r=12.5$$ $$2\pi r\times\frac{1}{2}=\pi r=\pi\times12.5$$ …


2012 June(70C) Math Question 53

Inequalities

- Graphing Inequalities

53. For all a>l, which of the following statements describes the function g defined by g(x)=a?

Choice C

Simply graph y = a-x …


Choice J

$$12^{2}f=2fd^{2}$$ $$144f=2fd^{2}$$ $$72=d^{2}$$ $$d=6\sqrt{2}$$


2012 June(70C) Math Question 55

Perimeter, Area, Volume

- Other Shapes

55. What is the area, in square meters, of a rhombus(a parallelogram having 4 congruent sides) with side length 4 meters and 2 angles that each measure 60°?

Choice A

70c

$$30°:60°:90°=1:\sqrt{3}:2$$ $$\Rightarrow 2:2\sqrt{3}:4$$ $$Area=2\times2\sqrt{3}\times\frac{1}{2}\times4$$ $$=8\sqrt{3}$$


Choice J

$$\because 13^{2}=5^{2}+12^{2}$$

The unknown leg …


Choice D

$$\frac{1}{11^{20}}-\frac{1}{11^{21}}$$ $$=\frac{1\times11}{11^{20}\times11}-\frac{1}{11^{21}}$$ $$=\frac{11}{11^{21}}-\frac{1}{11^{21}}$$ $$=\frac{10}{11^{21}}$$


2012 June(70C) Math Question 58

Trigonometry

- Pythagorean Theorem & sin cos tan

58. Which of the following is equal to tan theta cos theta when sin theta=2/3  and 0 < theta < pi/2?

Choice F

70c

$$\tan\theta\times\cos\theta$$ $$=\frac{2}{\sqrt{5}}\times\frac{\sqrt{5}}{3}$$ $$=\frac{2}{3}$$


Choice E

70c

The side length of each …


2012 June(70C) Math Question 60

Absolute Values

- Absolute Value Expressions

60. If  | x | = - x, which of the following statements must be true?

Choice F

For example, if x=-1 then …