ACT Form 69F Math Answer Explanation



Choice C

$$14+9=23$$ $$23\times2=46$$


2011 June(69F) Math Question 2

Sequences and Patterns

- Geometric Sequences

2. The Ist term in the geometric sequence below is-4. If it can be determined. what is the 6th term?
-4, 8, -16, 32, -64, ...

Choice F

$$\frac{8}{-4}=-2$$ $$\frac{-16}{8}=-2$$ $$-64\times(-2)=128$$


Choice A

$$(8+3-(-5))(3+(-5))$$ $$=(8+3+5)(3-5)$$ $$=16(-2)$$ $$=-32$$


2011 June(69F) Math Question 4

Word Problems

- Fixed Cost + Variable Cost

4. Every year on Inari's birthday, her uncle gives her $10, plus $2 for each year of her age. How old will Inari be when she receives $34 from her uncle on her birthday?

Choice G

$$\frac{34-10}{2}=12$$


Choice E

$$-2x+3=-21$$ $$-2x=-21-3=-24$$ $$x=\frac{-24}{-2}=12$$


2011 June(69F) Math Question 6

Statistics

- Mean, Median

6. What is the mean of the list of numbers below?
62,100,93,62,72,78,50,85,62,36

Choice J

$$62+100+93+62+72+78+50+85+62+36=700$$ $$\frac{700}{10}=70$$


Choice E

$$1\frac{2}{3}+\frac{3}{4}=1\frac{8}{12}+\frac{9}{12}$$ $$=1\frac{17}{12}$$ $$=2\frac{5}{12}$$


Choice H

69f

$$6:10=x:90$$ $$10x=6\times90=540$$ $$x=54$$


Choice C

$$$35\times4+0.375\times300$$ $$=$140+$112.5$$ $$=$252.5$$


2011 June(69F) Math Question 10

Quadratic Equations

- FOIL

10. The expression(7c- 2d)(2c+ d) is equivalent to

Choice J

$$=7c(2c+d)-2d(2c+d)$$ $$=14c^{2}+7cd-4dc-2d^{2}$$ $$=14c^{2}+3cd-2d^{2}$$


Choice B

$$3x+5y-13=0$$ $$5y=13-3x$$ $$y=\frac{13-3x}{5}$$


2011 June(69F) Math Question 12

Word Problems

- Distance, Speed, Time

12. The speed of one bus exceeds 3 times the speed of another bus by 20 mph. The speed of the slower bus is k mph. Which of the following expressions represents the speed of the faster bus, in miles per hour?

Choice K

“3 times the speed” of …


2011 June(69F) Math Question 13

Word Problems

- Temperature

13. At 2:00 a m, the temperature is -15°F. For the next
3 hours, the temperature decreases 2°F per hour. What is the temperature at 5:00 a m.?

Choice A

$$-15-2\times3=-15-6=-21$$


Choice J

From 2 hours to 4 hours, her distance traveled does not increase. Therefore, her car must have been stopped.

$$250-100=150$$


2011 June(69F) Math Question 15

Perimeter, Area, Volume

- Other Shapes

15. A children's wading pool, shown below, is in the shape of a right circular cylinder and has a diameter of 6 feet The pool is filled to a uniform depth of 1.5 feet. which of the following values is closest to the volume of water in the pool, in cubic feet?

Choice B

$$r=\frac{6}{2}=3$$ $$h=1.5$$ $$\pi \approx 3.14$$ $$\pi r^{2}h=3.14\times 3^{2}\times(1.5)=42.39$$


2011 June(69F) Math Question 16

Absolute Values

- Absolute Value Expressions

16. For the function f(x)=| x |3 -1, what is the value of f(-1)?

Choice F

$$f(-1)=\mid-1\mid^{3}-1=1^{3}-1=1-1=0$$


Choice C

$$332\times1.25=415$$


Choice F

$$6a-4a+5b-4b+4c+2c=2a+b+6c$$


2011 June(69F) Math Question 19

Equation of a Line

- Distance & Midpoint

19. In the standard (x,y) coordinate plane, the coordinates of the endpoints of GN are(6, 8)and(14, 18). what is the x-coordinate of the midpoint of GN?

Choice B

$$\frac{6+14}{2}=10$$


2011 June(69F) Math Question 20

Perimeter, Area, Volume

- Mixed Shapes

Use the following information to answer questions 20-22
In the figure shown below, each side of square ABCD is
6 inches long. Point E lies inside the square, forming the equilateral triangle △ECD.
20. What is the area, in square inches, of square ABCD?

Choice K

$$6\times6=36$$


2011 June(69F) Math Question 21

Trigonometry

- Pythagorean Theorem & sin cos tan

21. What is the measure of ∠BCE

Choice C

69f


Choice G

If “the x-coordinate of point …


2011 June(69F) Math Question 23

Word Problems

- % Concentration

23. Beverly is using a punch recipe that calls for 2 parts ginger ale to I part fruit juice by volume. Bever planning to make 12 liters of this punch for a party How many liters of fruit juice does Beverly need in order to make 12 liters of this punch?

Choice C

You must add these parts …


Choice F

$$\frac{4c}{5}-\frac{1}{2}=-\frac{25}{2}$$ $$\frac{4c}{5}=-\frac{25}{2}+\frac{1}{2}$$ $$\frac{4c}{5}=-\frac{24}{2}$$ $$4c\times2=5\times(-24)$$ $$8c=-120$$ …


Choice B

69f

$$\overline{AC}=r+4r=5r$$ $$\overline{AB}:\overline{AC}=r:5r=1:5$$


Choice F

$$y=mx+b$$ $$3y=-8x+5$$ $$y=-\frac{8}{3}x+5$$ $$m=-\frac{8}{3}$$


2011 June(69F) Math Question 27

Probability

- Combinations

27. The Kingston High School multimedia room has
108 seats, which are all arranged in rows. The number of seats in each row is 3 less than the number of rows How many rows of seats are in the multimedia room?

Choice C

Let r = number of …


2011 June(69F) Math Question 28

Trigonometry

- Graphing Trig Functions

28.If 90°< theta <180° and sin theta =10, then cos theta=?

Choice H

69f

$$90°\lt\theta\lt180°$$

It is in the …


Choice C

Because the length of the …


Choice G

$$1\times316+2\times271=858$$


Choice E

$$12d:30d:40d=64:24:18$$ $$[64,24,18]=576$$


Choice K

If the board is 1.5 …


2011 June(69F) Math Question 33

Conics

- Circles

33. In the standard (x, y) coordinate plane, the equation of a circle is x+y=100. At what 2 points does the circle intersect the y-axis?

Choice A

This equation is

$$(x-h)^{2}+(y-k)^{2}=r^{2}$$

where …


Choice H

$$(-mp)(mp^{9})^{4}$$ $$=(-m)(m^{4})(p)(p^{9\times4})$$ $$=(-m^{5})(p{37})$$


2011 June(69F) Math Question 35

Sequences and Patterns

- General Sequences

35. The fraction
578/999
is equivalent to the repeating decimal
0.578. What is the 85th digit to the right of the decimal point?

Choice A

$$0.578578578...$$

Because the decimal repeats …


2011 June(69F) Math Question 36

Probability

- More than One Event

36. A jar contains exactly 12 gumballs, each of which is a solid color. There are 5 blue and 7 white gumballs Chris will draw one gumball at random and then without replacement, will draw another gumball from the jar at random. Which of the following expressions gives the probability that Chris will draw 2 blue gumballs?

Choice K

His probability of drawing a …


Choice E

$$red=green+15,\ green=blue+9$$ $$red=blue+9+15=blue+24$$ $$red+green+blue=72$$ $$\Rightarrow …


2011 June(69F) Math Question 38

Perimeter, Area, Volume

- Other Shapes

38. A contractor charges s1.75 per square foot to waterproof the top surface of a wooden deck. What would be the contractor's charge for waterproofing the top surface of the trapezoid-shaped wooden deck shown below?

Choice H

$$(14+24)10\times\frac{1}{2}$$ $$=38\times10\times\frac{1}{2}$$ $$=380\times\frac{1}{2}$$ $$=190$$ $$190\times$1.75=$332.5$$


Choice A

The slope of OC is …


2011 June(69F) Math Question 40

Word Problems

- 2 Variable Addition

40. The junior class at Silverwood High School sponsored a costume party. Each student wearing a costume paid
2 admission, and each student NOT wearing a costume paid $5 admission. An amount of $489 was collected from the 150 students who paid admission.
How much of the amount was collected from students NOT wearing a costume?

Choice G

$$Pay\ $2:\ x$$ $$Pay\ $5:\ …


2011 June(69F) Math Question 41

Number Types & Properties

- Complex Number

41. Which of the following complex numbers is a sum of V-48 and y-27?

Choice D

$$i=\sqrt{-1},\ i^{2}=-1$$ $$\sqrt{-27}=\sqrt{27}i=3\sqrt{3}i$$ $$\sqrt{-48}=\sqrt{48}i=4\sqrt{3}i$$ $$\sqrt{-48}+\sqrt{-27}$$ …


2011 June(69F) Math Question 42

Trigonometry

- Pythagorean Theorem & sin cos tan

42. Point P lies on side AD of rectangle ABCD as shown below. The measure of ZABP is 67 and the length of AB is 7.5 inches. Which of the following expressions is the length, in inches, of AP

Choice K

$$tangent=\frac{opposite}{adjacent}$$ $$\tan67°=\frac{x}{7.5}$$ $$x=7.5\tan67°$$


2011 June(69F) Math Question 43

Angles

- Circles

43. For any circle with 2 secants drawn from P to the circle, as shown in the figure below, the lengths of the segments of the secants are determined by a(a+b)=c(c+d).
In the figure below, A, B, D, and E are on the circle Point B is on secant segment AC, and D is on secant segment CE. The lengths of AB, BC, and CD are given in meters. What is the length, in meters, of DE

Choice B

In the diagram,

$$a = …


2011 June(69F) Math Question 44

Angles

- Other Shapes

44. While lifting weights, a person sits on an incline weight bench and pushes the weights straight up, as shown in the figure below on the left. The weight bench is shown below on the right. The angle between the horizontal seat and the backrest is 110 what is the measure of the angle between the backrest and the vertical path of the weights?

Choice J

69f

This line then forms an …


2011 June(69F) Math Question 45

Conics

- Parabola

45. In the standard (x, y) coordinate plane, the graph of x is shifted 5 units down and 4 units right. Which of the following is an equation of the translated graph?

Choice D

To shift this graph down …


2011 June(69F) Math Question 46

Equation of a Line

- Distance & Midpoint

46. When Kaylee kicked the winning soccer goal, she was
10 yards behind the midfield line and 5 yards from the sideline, as shown in the figure below, To the nearest yard, how far was Kaylee from the center of the goal line when she kicked the winning goal?

Choice G

69f

$$\frac{60}{2}-5=25$$ $$\frac{100}{2}+10=60$$ $$25^{2}+60^{2}=625+3600=4225$$ $$4225=65^{2}$$


2011 June(69F) Math Question 47

Angles

- Circles

47. A sector of a circle has an area equal to 5/12 of the area of the circle. What is the measure of the sectors central angle?

Choice D

$$360°\times\frac{5}{12}=150°$$


2011 June(69F) Math Question 48

Statistics

- Mean, Median

48. List B consists of all the integers in List a below and also 3 integers a, b, and c, where a s 17, b=c, and b ≥ 42. what is the median of the integers in List B?
List A:12,16,17,29,29,35,41,42,47,4

Choice J

69f

$$\frac{35+41}{2}=38$$


2011 June(69F) Math Question 49

Absolute Values

- Absolute Value Expressions

49. The solution set for the inequality | 2x-a | ≤ 3 is  { x  |1≤x≤4l} . What is the value of a?

Choice D

$$2x-a\leq3 \Rightarrow x\leq\frac{3+a}{2}$$ $$2x-a\geq-3 \Rightarrow …


Choice J

The greatest power of 17,23,61 …


2011 June(69F) Math Question 51

Trigonometry

- Pythagorean Theorem & sin cos tan

51. As shown in the figure below, the drill holes for Well A and Well B are 300 feet apart on level ground Well A is dug straight down and reaches oil at
2,000 feet. Well B is dug straight down and reaches oi at 1,500 feet. What is the angle of elevation from the point where Well A reaches oil to the point where Well B reaches oil?
(Note:The figure is Not drawn to scale.)

Choice E

Based on the diagram, the …


Choice G

69f

$$f(5)=3$$ $$g(f(5))=g(3)=-5$$


2011 June(69F) Math Question 53

Trigonometry

- Pythagorean Theorem & sin cos tan

53. In the figure below, △ACD is a right triangle, b lies on AC, and the measures of ∠ DAC and  ∠ DBC are given The length of AB is 10 yards. what is the approximate length, in yards, of CD?

Choice C

69f

$$45°:45°:90°$$ $$=1:1:\sqrt{2}$$ $$=x:x:x\sqrt{2}$$ $$30°:60°:90°$$ $$=1:\sqrt{3}:2$$ …


Choice K

$$a+b=5$$ $$ab=6$$ $$a,b=2,3$$ $$\Rightarrow \frac{3}{2}+\frac{2}{3}=\frac{9+4}{6}=\frac{13}{6}$$


2011 June(69F) Math Question 55

Matrices

- Addition & Subtraction

55. Given the matrix equation shown below, what is = b/a?
(Note: Whenever n is a positive integer, the notation n!
represents the product of the integers from n to 1. For example,3!=3·2·1)

Choice A

$$a= 3! + 2! = …


2011 June(69F) Math Question 56

Inequalities

- Intersection of Lines

56. Graphed in the standard (x, coordinate plane below is the solution set for a system of 3 linear inequalities For values of x and y that satisfy this system, what is the maximum value of 6x- 2y?

Choice K

To find the maximum value …


2011 June(69F) Math Question 57

Perimeter, Area, Volume

- Other Shapes

57. The square base of a regular pyramid has a side length of 4 inches. Each of the other 4 faces of the pyramid is a triangle with a base of 4 inches and a height of 6 inches. The pyramid has a total surface area of 64 square inches. A second regular pyramid has a square base that is 4 inches by 4 inches, but its total surface area is double that of the first pyramid. what is the height, in inches, of each of the triangular faces of the second pyramid?

Choice C

If the second pyramid has …


2011 June(69F) Math Question 58

Trigonometry

- Advanced sin, cos, tan functions

58. As shown in the figure below, a triangle has 2 sides each of length 5 feet and a 3rd side of length x feet.
The degree measure of the angle between the 2 sides that are 5 feet long is 8. In terms of x, cos 0=?
(Note: For any triangle, if a, b, and c are the lengths of the sides opposite ∠A,  ∠B, and ∠C, respectively, then a2=b2+c2-2 bc cos ∠A)

Choice G

$$x^{2}=5^{2}+5^{2}-2\cdot5\cdot5\cos\theta$$ $$x^{2}=25+25-50\cos\theta$$ $$50\cos\theta=50-x^{2}$$ $$\cos\theta=\frac{50-x^{2}}{50}$$


Choice D

SAS Triangle Congruence Theorem : …


2011 June(69F) Math Question 60

Quadratic Equations

- Factoring

60. What is the value of c if x+2 is a factor of
2x3+2x2-2cx+4?

Choice J

If x + 2 is …