ACT Form 66F Math Answer Explanation



2009 April(66F) Math Question 1

Trigonometry

- Pythagorean Theorem & sin cos tan

1. On level ground, a vertical rod 12 feet tall casts a shadow 4 feet long, and at the same time a nearby vertical flagpole casts a shadow 12 feet long. How many feet tall is the flagpole?

Choice E

$$12:4=x:12$$ $$4x=144$$ $$x=36$$


2009 April(66F) Math Question 2

Word Problems

- Fixed Cost + Variable Cost

2. The cost of a gym membership is a onetime fee of $140, plus a monthly fee of $40. Brendan wrote a S500 check to pay his gym membership for a certain number of months, including the onetime fee. How
,many months of membership did he pay for?

Choice H

$$500-140=360$$ $$360\div40=9$$


Choice A

$$\frac{x^{2}-1}{x+1}$$ $$=\frac{(-5)^{2}-1}{(-5)+1}$$ $$=\frac{24}{-4}$$ $$=-6$$


2009 April(66F) Math Question 4

Word Problems

- 1 Variable Equations

4. A museum offers a 2-hour guided group tour. For groups with fewer than
25 people the cost is $9. 25 person: for groups with 25 people or more the cost is
350 per person. The 27 people in the 9:00 a.m. tour group each paid $9.25 in advance. What is the total refund that the museum owes the 9:00 a.m. group?

Choice J

$$9.25-8.5=0.75$$ $$0.75\times27=20.25$$


2009 April(66F) Math Question 5

Probability

- One Event

5. The 13-member math club needs to choose a student government representative. They 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 Samara, who is a member of the club but NOT an officer, will be chosen?

Choice C

$$P=\frac{1}{13-3}=\frac{1}{10}$$


Choice H

$$(15+6)\times2=42$$


2009 April(66F) Math Question 7

Word Problems

- 2 Variable Addition

7. Tickets for a community theater production cost $6
each when bought in advance and S8 each when bought at the door. The theater group's goal is at least
$2,000 in ticket sales for opening night. The theater group sold 142 opening-night tickets in advance. What is the minimum number of tickets they need to sell at the door on opening night to make their goal?

Choice B

$$6\times142=852$$ $$2000-852=1148$$ $$1184\div8=143.5$$


Choice J

$$\frac{8}{12}=\frac{10}{r}$$ $$8r=120$$ $$r=15$$


Choice E

$$12(x-11)=-15$$ $$\Rightarrow 12x-132=-15$$ $$\Rightarrow 12x=117$$ $$\Rightarrow 4x=39$$ $$\Rightarrow x=\frac{39}{4}$$


2009 April(66F) Math Question 10

Angles

- Collinear Points & Triangles

10. In the figure below, A, D, C, and E are collinear. AD, BD, and BC are all the same length, and the angle measure of ∠ABD is as marked. What is the degree measure of ∠BCE?


Choice C

$$f(-2)=9(-2)^{2}+5(-2)-8$$ $$=9(4)-10-8$$ $$=36-18$$ $$=18$$


Choice J

$$30=3\times10$$ $$20=2\times10$$ $$70=7\times10$$ $$[30,20,70]=2\times3\times7\times10=420$$


Choice E

$$Originally\ wanted=\frac{x}{2}$$ $$(x\times2)\div\frac{x}{2}=4$$ $$\Rightarrow (x\times2)\div4=\frac{x}{2}$$


2009 April(66F) Math Question 14

Perimeter, Area, Volume

- Squares in a Shape

14. The 8-sided figure below is divided into 5 congruent squares. The total area of the 5 squares is 125 square inches. What is the perimeter, in inches, of the figure?

Choice G

66f

$$125\div5=25$$ $$25=5\times5$$ $$5\times12=60$$


Choice E

$$\angle B=90°$$ $$\angle A+\angle B+\angle C=180°$$ $$\Rightarrow \angle A+\angle C=90°$$ $$\because \angle A\gt 43°$$ $$90-43=47$$ $$\therefore \angle C \lt 47°$$


Choice G

$$I. $$ $$(2+1)^{4}+(6+3)^{2}\neq10$$ $$II. $$ $$(2-1)^{4}+(6-3)^{2}$$ $$=(1)^{4}+(3)^{2}$$ $$=1+9$$ $$=10$$ $$III.$$ $$(2\div1)^{4}+(6\div3)^{2}$$ $$=(1)^{4}+(2)^{2}$$ $$=1+4$$ $$=5$$ $$\neq 10$$


Choice E

66f

$$(0,5)\Rightarrow 5=2(0)+5$$ $$Only\ (E)\ is\ correct.$$


Choice F

$$\Rightarrow (n+4)8=2n-10$$


2009 April(66F) Math Question 19

Word Problems

- Fixed Cost + Variable Cost

19. Two workers were hired to begin work at the same time, Worker A's contract called for a starting salary of $20,000 with an increase of $800 after each year of employment. Worker B's contract called for a starting salary of $15, 200 with an increase of $2,000 after each year of employment. If x represents the number of full years' employment (that is, the number of yearly increases each worker has received), which of the following equations could be solved to determine the number of years until Bs yearly salary equals A's yearly salary?

Choice A

Assuming that x years have passed...

$$\Rightarrow 20,000+800x=15,200+2,000x$$


2009 April(66F) Math Question 20

Trigonometry

- Pythagorean Theorem & sin cos tan

20. A ramp for loading trucks is 13 feet long and covers
12 feet along the level ground, as shown below. How many feet high is the highest point on the ramp?

Choice J

66f

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


Choice E

$$=7x+21-6x+6$$ $$=x+27$$


Choice G

$$x+y=32---(1)$$ $$x-y=12---(2)$$ $$(1)+(2)$$ $$\Rightarrow 2x=44$$ …


2009 April(66F) Math Question 23

Quadratic Equations

- FOIL

23. When(2x-3)^2 is written in the form ax² + bx +c, where a, b, and c are integers, a+b+c?

Choice C

$$(2x-3)^{2}$$ $$=4x^{2}-12x+9$$ $$\Rightarrow a=4,b=-12,c=9$$ $$a+b+c=4-12+9=1$$


Choice H

66f

$$\Rightarrow 15\times15+5\times10$$ $$=225+50$$ $$=275$$


Choice A

66f

It can be seen from the table that f(x) moves in increments of 0.2, meaning f(x) is a linear function. It follows that f(4) = 0.2.


Choice H

$$y=mx+b$$ $$6y=14x+5$$ $$y=\frac{14}{6}+\frac{5}{6}$$ $$m=\frac{14}{6}=\frac{7}{3}$$


2009 April(66F) Math Question 27

Quadratic Equations

- Factoring

27. What is the sum of the 2 solutions of the equation x² + x -12=0?

Choice C

$$x^{2}+x-12$$ $$=(x-3)(x+4)$$ $$\Rightarrow x=3,-4$$ $$3+(-4)=-1$$


Choice J

$$3+5+7=15$$ $$\Rightarrow 15\times\frac{5}{3}=25$$


2009 April(66F) Math Question 29

Statistics

- Mean, Median

29. At a certain location, the low temperatures. in degrees Fahrenheit, for each of 7 consecutive days in January were-2°F, 4°F, -3°F, 1°F, 2°F, -5°F, and -6°F. What was the median of these low temperatures?

Choice A

$$\Rightarrow -6,-5,-3,-2,1,2,4$$ $$The\ median:\ -2$$


2009 April(66F) Math Question 30

Quadratic Equations

- Factoring

30. When asked his age, the algebra teacher said, If you square my age, then subtract 23 times my age, the result is How old is he?

Choice G

$$x^{2}-23(x)=50$$ $$\Rightarrow x^{2}-23(x)-50=0$$ $$\Rightarrow (x-25)(x+2)=0$$ …


2009 April(66F) Math Question 31

Word Problems

- Distance, Speed, Time

31. The distance, d, an accelerating object travels in r seconds can be modeled by the equation d=½at^2 where a is the acceleration rate, in meters per second per second. If a car accelerates from a stop at the rate of 20 meters per second per second and travels a distance of 80 meters, about how many seconds did the car travel?

Choice B

$$d=\frac{1}{2}at^{2}$$ $$80=\frac{1}{2}(20)t^{2}$$ $$80=10t^{2}$$ $$8=t^{2}$$ $$t\approx2.82$$


2009 April(66F) Math Question 32

Sequences and Patterns

- Geometric Sequences

32. Let a, b, c, and d be distinct positive integers. What i the 4th term of the geometric sequence below?
bcd, abcd, abcd, …

Choice F

$$\frac{abc^{2}d}{bcd}=ac$$ $$\frac{a^{2}bc^{3}d}{abc^{2}d}=ac$$ $$\Rightarrow a^{2}bc^{3}d(ac)=a^{3}bc^{4}d$$


2009 April(66F) Math Question 33

Probability

- One Event

Use the following information to answer questions 33-35.
A survey in a study skills class asked the 20 students enrolled in the class how many hours (rounded to the nearest hour) they had spent studying on the previous evening. The 20 responses are summarized by the histogram below.
33. What fraction of the students responded that they had spent less than 3 hours studying?

Choice D

66f

$$\Rightarrow \frac{2+5+6}{20}$$ $$=\frac{13}{20}$$


2009 April(66F) Math Question 34

Angles

- Circles

34. The teacher decides to show the data in a circle graph
(pie chart). What should be the measure of the central angle of the sector for 3 hours?

Choice J

66f

$$\Rightarrow \frac{4}{20}=\frac{2}{10}$$ $$\frac{2}{10}\times360°=72°$$


2009 April(66F) Math Question 35

Statistics

- Mean, Median

35. To the nearest tenth of an hour, what is the average number of hours for the 20 survey responses?

Choice B

$$=\frac{0(2)+1(5)+2(6)+3(4)+4(2)+5(1)}{20}$$ $$=\frac{0+5+12+12+8+5}{20}$$ $$=\frac{42}{20}$$ $$=2.1$$


2009 April(66F) Math Question 36

Quadratic Equations

- Factoring

36. For all x > 21 ,  (x²+8x+7)(x-3) / (x²+4x-21)(x+1) =?

Choice F

$$\Rightarrow\frac{(x+7)(x+1)}{(x+7)(x-3)}\times\frac{x-3}{x+1}$$ $$\Rightarrow 1$$


2009 April(66F) Math Question 37

Trigonometry

- Pythagorean Theorem & sin cos tan

37. The bottom of the basket of a hot-air balloon is parallel to the level ground. One taut tether line 144 feet long is attached to the center of the bottom of the basket and is anchored to the ground at an angle of 72, as shown in the figure below. Which of the following expressions gives the distance, in feet, from the center of the bottom of the basket to the ground?

Choice E

66f

$$\sin 72°=\frac{x}{144}$$ $$x=144\sin 72°$$


2009 April(66F) Math Question 38

Equation of a Line

- Distance & Midpoint

38. The coordinates of the endpoints of GH. in the standard (x, y) coordinate plane, are(-8, -3)and(2, 3). What is the x- coordinate of the midpoint of GH?

Choice G

$$\Rightarrow(\frac{-8+2}{2},\frac{-3+3}{2})$$ $$=(\frac{-6}{2},\frac{0}{2})$$ $$=(-3,0)$$ $$\Rightarrow x=-3$$


2009 April(66F) Math Question 39

Equation of a Line

- Distance & Midpoint

39. On a map in the standard (x, y) coordinate plane, the towns of Arlington and Betelwood are represented by the points(-2, -3)and(-6,-7), respectively. Each unit on the map represents an actual distance of 10 miles.
Which of the following is closest to the distance, in miles, between these 2 towns?

Choice B

$$\Rightarrow\sqrt{(-6-(-2))^{2}+(-7-(-3))^{2}}$$ $$=\sqrt{16+16}$$ $$=\sqrt{32}$$ $$\approx 5.65$$ …


Choice K

"The sum of two rational …


2009 April(66F) Math Question 41

Number Types & Properties

- Complex Number

41. For the complex number i and an integer x, which of the following is a possible value of i?

Choice B

$$i=\sqrt{-1}$$ $$i^{2}=-1$$ $$\Rightarrow (i^{2})^{2}=1$$ $$\Rightarrow …


Choice H

$$x=3$$ $$y=(3-3)^{2}+2=2$$


2009 April(66F) Math Question 43

Perimeter, Area, Volume

- Other Shapes

43. The height and radius of the right circular cylinder below are given in meters. What is the volume, in cubic meters. of the cylinder?

Choice C

$$\pi r^{2}=25\pi$$ $$25\pi\times6=150$$


Choice F

66f

$$The\ perimeter\ of\ \triangle ABC:\triangle …


2009 April(66F) Math Question 45

Trigonometry

- Pythagorean Theorem & sin cos tan

45.A rocket lifted off from a launch pad and traveled vertically 30 kilometers, then traveled 40 kilometers at 30 from the vertical, and then traveled 100 kilometers at 45 from the vertical, as shown in the figure below At that point, the rocket was how many kilometers above the height of the launch pad?

Choice E

66f

$$\triangle 45°-45°-90°=1:1:\sqrt{2}$$ $$\triangle 30°-60°-90°=1:\sqrt{3}:2$$ $$H=\frac{100}{\sqrt{2}}+20\sqrt{3}+30$$ …


2009 April(66F) Math Question 46

Conics

- Circles

46. In the standard (x, y) coordinate plane, what is the area of the circle x +y=16?

Choice H

The equation of a circle …


2009 April(66F) Math Question 47

Perimeter, Area, Volume

- Mixed Shapes

47. In the standard (x, y) coordinate plane below, I side of a rectangle is on the x-axis, and the vertices of the opposite side of the rectangle are on the graph of the parabola y =6-2
Let a represent any value of x such that o < x < √3. Which of the following is an expression in terms of a for the area, in square coordinate units, of any such rectangle?

Choice A

$$2a\times(6-2a^{2})$$ $$=12a-4a^{3}$$


Choice F

$$Let n be 2 or …


2009 April(66F) Math Question 49

Logarithms

- Logarithm Expressions

49. The value of log5(5^13/2)  is between which of the following pairs of consecutive integers?

Choice D

Logarithm base 5 of 5 …


Choice J

$$\frac{156-144}{144}=\frac{12}{144}\approx0.083$$


2009 April(66F) Math Question 51

Statistics

- Mean, Median

51. Prior to Week 3, the manager set the goal for meat department sales to average 150 thousand dollars per week from the beginning of Week 3 through the end of Week 6. To meet this goal, what must be the minimum meat department sales for Week 6, in thousands of dollars?

Choice D

$$\frac{156+145+140+x}{4}=150$$ $$441+x=600$$ $$x=159$$


Choice G

$$Week\ 1=30$$ $$Week\ 2=30(1.1)=33$$ $$Week\ …


2009 April(66F) Math Question 53

Trigonometry

- Graphing Trig Functions

53. A trigonometric function with equation y= a sin(bx +c)。
where a, b, and c are real numbers, is graphed in the standard (x, y) coordinate plane below. The period of this function f(x)is the smallest positive number p such that f(x +p)=f(x) for every real number x. One of the following is the period of this function. which one is it?

Choice B

66f


2009 April(66F) Math Question 54

Trigonometry

- Pythagorean Theorem & sin cos tan

54. The building below casts a shadow 24 yards long, and the angle of elevation from the tip of the shadow to the top of the building has a sine of 4/5 . What is the height of the building in yards?

Choice K

66f

$$3:4:5$$ $$=24:32:40$$ $$The\ height=32$$


2009 April(66F) Math Question 55

Absolute Values

- Absolute Value Expressions

55. Which of the following is an irrational number that is a solution to the equation | x² - 12 | - 4 = 0?

Choice C

$$\mid x^{2}-12 \mid -4=0$$ $$\Rightarrow\ …


2009 April(66F) Math Question 56

Graphing Properties

- Interpret Graphs or Figures

56. The graph below shows the distance Malika is from a motion detector for a period of 10 seconds. A certain order of 3 of the following 5 actions describes Malika's movement in relation to the position of the motion detector. Which order is it?

Choice K

66f

$$10\rightarrow2=Walk\ toward$$ $$Speed=\frac{8}{4}=2$$ $$2\rightarrow6=Walk\ away$$ …


2009 April(66F) Math Question 57

Angles

- Circles

57. As shown in the figure below, a clock has a minute hand that measures 5 cm from its tip to the center of the clock. To the nearest centimeter. what is the distance traveled by the tip of the minute hand between 2:10 p.m. and 5:30 p.m.?

Choice D

$$2:10\rightarrow 5:30$$ $$=3\ hours\ 20mins=3\frac{1}{3}\ …


2009 April(66F) Math Question 58

Trigonometry

- Pythagorean Theorem & sin cos tan

58. For an angle whose measure is between 90° and 180°
tan◆ = -7/24
Which of the following equals sin◆ ?

Choice K

66f

$$24^{2}+7^{2}=25^{2}$$ $$\Rightarrow \sin \phi=\frac{7}{25}$$


Choice B

$$w+z=5$$ $$(w,z)=(1,4),(2,3),(3,2),(4,1)$$ $$2^{w}=x$$ $$x^{z}=64$$ $$(1,4)\Rightarrow2^{1}=2,\ …


2009 April(66F) Math Question 60

Perimeter, Area, Volume

- Mixed Shapes

60. A cube with edges 4 cm long is inscribed in a sphere as shown below. What is the radius of the sphere, in centimeters?

Choice F

66f

$$\sqrt{(4\sqrt{2})^{2}+4^{2}}=\sqrt{48}=4\sqrt{3}$$ $$4\sqrt{3}=2r$$ $$r=2\sqrt{3}$$