ACT Past Papers with Answers

Paper 71

ACT April 2007

Form 64E · Math Test · 60 questions

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Paper 71 · ACT April 2007 · Form 64E · Math Test

⏐7 − 3⏐ − ⏐3 − 7⏐ = ?

A−8B−6C−4D0E8

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Paper 71 · ACT April 2007 · Form 64E · Math Test

A consultant charges $45 for each hour she works on a consultation, plus a flat $30 consulting fee. How many hours of work are included in a $210 bill for a consultation?

A2B4C4D5E7

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Vehicle A averages 14 miles per gallon of gasoline, and Vehicle B averages 36 miles per gallon of gasoline. At these rates, how many more gallons of gasoline does Vehicle A need than Vehicle B to make a 1,008-mile trip?

A25B28C44D50E72

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Paper 71 · ACT April 2007 · Form 64E · Math Test

t2 − 59t + 54 − 82t2 + 60t is equivalent to: + t2 + 54 + t + 54 + t + 54

A−26t2B−26t6C−81t4D−81t2E−82t2

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Paper 71 · ACT April 2007 · Form 64E · Math Test

The figure below is composed of square BCDE and equilateral triangle 䉭ABE. The length of CD ___ is 6 inches. What is the perimeter of ABCDE, in inches?

A18B24C30D42E45

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Paper 71 · ACT April 2007 · Form 64E · Math Test

The expression (4z + 3)(z − 2) is equivalent to: − 5 − 6 − 3z − 5 − 5z − 6 + 5z − 6

A4z2B4z2C4z2D4z2E4z2

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Paper 71 · ACT April 2007 · Form 64E · Math Test

If 40% of a given number is 8, then what is 15% of the given number?

A1.2B1.8C3.0D5.0E6.5

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Paper 71 · ACT April 2007 · Form 64E · Math Test

The 6 consecutive integers below add up to 447. x − 2 x − 1 x x + 1 x + 2 x + 3 What is the value of x ? A B C D E 6 1 _ _ 2 2 _ _ 3 4 _ _ 5 GO ON TO THE NEXT PAGE.

A72B73C74D75E76

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the standard (x,y) coordinate plane, point M with coordinates (5,4) is the midpoint of AB ___ , and B has coordinates (7,3). What are the coordinates of A ?

A(17,11)B( 9, 2)C( 6, 3.5)D( 3, 5)E(−3,−5)

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Rectangle ABCD has vertices A(4,5), B(0,2), and C(6,−6). These vertices are graphed below in the standard (x,y) coordinate plane. What are the coordinates of vertex D ?

A(10,−3)B( 9,−2)C( 8, 2)D( 7, 1)E( 2,−9)

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Daisun owns 2 sportswear stores (X and Y). She stocks 3 brands of T-shirts (A, B, and C) in each store. The matrices below show the numbers of each type of T-shirt in each store and the cost for each type of T-shirt. The value of Daisun’s T-shirt inventory is computed using the costs listed. What is the total value of the T-shirt inventory for Daisun’s 2 stores? A B C Cost X 冤100 200 150 冥 A 冤 $ 5 冥 Y 120 50 100 B $10 C $15

A$2,200B$2,220C$4,965D$5,450E$7,350

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Given the triangle shown below with exterior angles that measure x°, y°, and z° as shown, what is the sum of x, y, and z ? A poll of 200 registered voters was taken before the election for mayor of Springdale. All 200 voters indicated which 1 of the 4 candidates they would vote for. The results of the poll are given in the table below. Candidate Number of voters Blackcloud 50 Lue 80 Gomez 40 Whitney 30

A180B231C309D360ECannot be determined from the given information

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Paper 71 · ACT April 2007 · Form 64E · Math Test

What percent of the voters polled chose Whitney in the poll?

A15%B20%C25%D30%E40%

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Paper 71 · ACT April 2007 · Form 64E · Math Test

If the poll is indicative of how the 10,000 registered voters of Springdale will actually vote in the election, which of the following is the best estimate of the number of votes Lue will receive in the election?

A1,500B2,500C4,000D5,000E8,000

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Paper 71 · ACT April 2007 · Form 64E · Math Test

If the information in the table were converted into a circle graph (pie chart), then the central angle of the sector for Gomez would measure how many degrees?

A54°B72°C90°D108°E144°

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In square ABCE shown below, D is the midpoint of CE ___ . Which of the following is the ratio of the area of 䉭ADE to the area of 䉭ADB ? C D E B 72° 57° y° x° z° y x –2 O 6 4 2 –2 –4 –6 –8 –10 2 4 6 8 10 GO ON TO THE NEXT PAGE. 2 2 ACT-64E-PRACTICE 27 Use the following information to answer questions 13–15.

A1:1B1:2C1:3D1:4E1:8 A

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Which of the following is the slope of a line parallel to the line y = x − 4 in the standard (x,y) coordinate plane?

Source page for act-april-2007-form-64e-math-017

A−4B−C2DChoice D — see source imageEChoice E — see source image

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Janelle cut a board 30 feet long into 2 pieces. The ratio of the lengths of the 2 pieces is 2:3. What is the length, to the nearest foot, of the shorter piece?

A5B6C12D15E18

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Paper 71 · ACT April 2007 · Form 64E · Math Test

What is the smallest integer greater than 公僓僓僓 58 ?

A4B7C8D10E30

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Sergio plans to paint the 4 walls of his room with 1 coat of paint. The walls are rectangular, and, according to his measurements, each wall is 10 feet by 15 feet. He will not need to paint the single 3-foot-by-5-foot rectangular window in his room and the 3 -foot-by- 7-foot rectangular door. Sergio knows that each gallon of paint covers between 300 and 350 square feet. If only 1-gallon cans of paint are available, which of the following is the minimum number of cans of paint Sergio needs to buy to paint his walls?

A1B2C3D4E5

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Paper 71 · ACT April 2007 · Form 64E · Math Test

What values of x are solutions for x2 + 2x = 8 ?

A−4 and 2B−2 and 0C−2 and 4D0 and 2E6 and 8

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Paper 71 · ACT April 2007 · Form 64E · Math Test

For all a > 1, the expression equals:

Source page for act-april-2007-form-64e-math-022

AChoice F — see source imageB−a2Ca2D−EChoice K — see source image

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Paper 71 · ACT April 2007 · Form 64E · Math Test

If point M has a nonzero x-coordinate and a nonzero y-coordinate and the coordinates have opposite signs, then point M must be located in which of the 4 quadrants labeled below?

AI onlyBIII onlyCI or III onlyDI or IV onlyEII or IV only

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Paper 71 · ACT April 2007 · Form 64E · Math Test

The fixed costs of manufacturing basketballs in a factory are $1,400.00 per day. The variable costs are $5.25 per basketball. Which of the following expressions can be used to model the cost of manufacturing b basketballs in 1 day? y x II I III IV O quadrants of the standard (x,y) coordinate plane 1 __ a2 1 __ a2 1 _ _ 2 3a4 ___ _ 3a6 1 _ _ 2 2 _ _ 3 3 _ _ 2 3 _ _ 2 2 _ _ 3 GO ON TO THE NEXT PAGE. 2 2 ACT-64E-PRACTICE 28

A$1,405.25bB$5.25b − $1,400.00C$1,400.00b + $5.25D$1,400.00 − $5.25bE$1,400.00 + $5.25b

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the figure below, where 䉭ABC ∼ 䉭KLM, lengths given are in centimeters. What is the perimeter, in centimeters, of 䉭ABC ? (Note: The symbol ∼ means “is similar to.”)

A12B14C21D35E71

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Paper 71 · ACT April 2007 · Form 64E · Math Test

If = is true, then a = ?

A1B公僓 7C7D21E49

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Paper 71 · ACT April 2007 · Form 64E · Math Test

A hot-air balloon 70 meters above the ground is falling at a constant rate of 6 meters per second while another hot-air balloon 10 meters above the ground is rising at a constant rate of 15 meters per second. To the nearest tenth of a second, after how many seconds will the 2 balloons be the same height above the ground?

A8.9B6.7C2.9D0.4E0.2

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Paper 71 · ACT April 2007 · Form 64E · Math Test

A hiking group will go from a certain town to a certain village by van on 1 of 4 roads, from the village to a waterfall by riding bicycles on 1 of 2 bicycle paths, and then from the waterfall to their campsite by hiking on 1 of 6 trails. How many routes are possible for the hiking group to go from the town to the village to the waterfall to their campsite?

A6B12C24D48E220

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Cube A has an edge length of 2 inches. Cube B has an edge length double that of Cube A. What is the volume, in cubic inches, of Cube B ?

A4B8C16D32E64

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Paper 71 · ACT April 2007 · Form 64E · Math Test

A formula used to compute the current value of a savings account is A = P(1 + r)n , where A is the current value; P is the amount deposited; r is the rate of interest for 1 compounding period, expressed as a decimal; and n is the number of compounding periods. Which of the following is closest to the value of a savings account after 5 years if $10,000 is deposited at 4% annual interest compounded yearly?

A$10,400B$12,167C$42,000D$52,000E$53,782

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Paper 71 · ACT April 2007 · Form 64E · Math Test

A right circular cylinder is shown in the figure below, with dimensions given in centimeters. What is the total surface area of this cylinder, in square centimeters? (Note: The total surface area of a cylinder is given by 2πr2 + 2πrh where r is the radius and h is the height.)

A,300πB,400πC,500πD,600πE1,600π

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Given f (x) = 4x + 1 and g(x) = x2 − 2, which of the following is an expression for f 冸g(x)冹 ? + 4x + 1 + 4x − 1 − 7 − 1 + 8x − 1 20 20 3公僓 7 ___ ___ _ 7 3公僓 7 ___ ___ _ a公僓 7 3 _ _ 4 1 _ _ 2 15 L M K 12.5 B C A 3 GO ON TO THE NEXT PAGE. 2 2 ACT-64E-PRACTICE 29

A−x2Bx2C4x2D4x2E16x2 7.5

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Paper 71 · ACT April 2007 · Form 64E · Math Test

The table below shows the total number of goals scored in each of 43 soccer matches in a regional tournament. What is the average number of goals scored per match, to the nearest 0.1 goal? Total number of Number of matches goals in a match with this total 0 4 1 10 2 5 3 9 4 7 5 5 6 1 7 2

A1.0B2.8C3.0D6.1E17.1

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Lines a, b, c, and d are shown below and a 储 b. Which of the following is the set of all angles that must be supplementary to ∠x ?

A{1, 2}B{1, 2, 5, 6}C{1, 2, 9, 10}D{1, 2, 5, 6, 9, 10}E{1, 2, 5, 6, 9, 10, 13, 14}

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Paper 71 · ACT April 2007 · Form 64E · Math Test

(3x3 )3 is equivalent to:

AxB9x6C9x9D27x6E27x9

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Which of the following is equivalent to the inequality 4x − 8 > 8x + 16 ?

Ax < −6Bx > −6Cx < −2Dx > 2Ex < 6

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Paper 71 · ACT April 2007 · Form 64E · Math Test

As shown in the standard (x,y) coordinate plane below, P(6,6) lies on the circle with center (2,3) and radius 5 coordinate units. What are the coordinates of the image of P after the circle is rotated 90° clockwise ( ) about the center of the circle?

A(2, 3)B(3, 2)C(5,−1)D(6, 0)E(7, 3)

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Paper 71 · ACT April 2007 · Form 64E · Math Test

For right triangle 䉭KLM below, what is sin,∠M ?

Source page for act-april-2007-form-64e-math-038

AChoice F — see source imageBChoice G — see source imageCChoice H — see source imageDChoice J — see source imageEChoice K — see source image

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the figure below, B lies on AC ___ , BD ___ bisects ∠ABE, and BE ___ bisects ∠CBD. What is the measure of ∠DBE ?

A90°B60°C45°D30°ECannot be determined from the given information

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Paper 71 · ACT April 2007 · Form 64E · Math Test

If there are 8 × 1012 hydrogen molecules in a volume of 4 × 104 cubic centimeters, what is the average number of hydrogen molecules per cubic centimeter? A C D E B 公僓僓 44 ____ _ 12 10 ____ _ 公僓僓 44 公僓僓 44 ____ _ 10 12 _ __ 10 10 _ __ 12 12 cm 10 cm M L K 8 6 4 2 (2,3) P(6,6) –4 –2 2 4 6 8 x y –2 O x 8 9 10 11 1 2 3 a b c d 4 5 12 13 6 7 14 15 GO ON TO THE NEXT PAGE. 2 2 ACT-64E-PRACTICE 30

A5 × 10−9B2 × 103C2 × 108D32 × 1016E32 × 1048

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the figure below, a radar screen shows 2 ships. Ship A is located at a distance of 20 nautical miles and bearing 170°, and Ship B is located at a distance of 30 nautical miles and bearing 300°. Which of the following is an expression for the straight-line distance, in nautical miles, between the 2 ships? (Note: For 䉭ABC with side of length a opposite ∠A, side of length b opposite ∠B, and side of length c opposite ∠C, the law of cosines states c2 = a2 + b2 − 2ab cos,∠C.)

Source page for act-april-2007-form-64e-math-041

AChoice A — see source imageBChoice B — see source imageCChoice C — see source imageDChoice D — see source imageEChoice E — see source image

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Paper 71 · ACT April 2007 · Form 64E · Math Test

What rational number is halfway between and ?

Source page for act-april-2007-form-64e-math-042

AChoice F — see source imageBChoice G — see source imageCChoice H — see source imageDChoice J — see source imageEChoice K — see source image

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In isosceles trapezoid ABCD, AB ___ is parallel to DC ___ , ∠BDC measures 25°, and ∠BCA measures 35°. What is the measure of ∠DBC ?

A85°B95°C105°D115°E125°

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the figure below, the area of the larger square is 50 square centimeters and the area of the smaller square is 18 square centimeters. What is x, in centimeters?

A2B2公僓 2C4公僓 2D16E32

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Which of the following is a rational number?

Source page for act-april-2007-form-64e-math-045

A公僓 2B公僓 πC公僓 7DChoice D — see source imageEChoice E — see source image

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Paper 71 · ACT April 2007 · Form 64E · Math Test

If a < b, then ⏐a − b⏐ is equivalent to which of the following?

Source page for act-april-2007-form-64e-math-046

Aa + bB−(a + b)CChoice H — see source imageDa − bE−(a − b)

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Tom has taken 5 of the 8 equally weighted tests in his U.S. History class this semester, and he has an average score of exactly 78.0 points. How many points does he need to earn on the 6th test to bring his average score up to exactly 80.0 points? 公僓僓僓僓僓 a − b 64 __ _ 49 5 __ _ 25 x C D B A 25° 35° 8 _ __ 15 4 _ __ 15 2 _ __ 15 1 _ _ 4 1 _ _ 2 1 _ _ 3 1 _ _ 5 公僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓 202 + 302 − 2(20)(30)cos470° 公僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓 202 + 302 − 2(20)(30)cos300° 公僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓 202 + 302 − 2(20)(30)cos170° 公僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓 202 + 302 − 2(20)(30)cos130° 公僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓僓 202 + 302 − 2(20)(30)cos60° B 170° 30 A N 20 300° ? GO ON TO THE NEXT PAGE. 2 2 ACT-64E-PRACTICE 31

A90B88C82D80E79

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the complex plane, the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. The complex number a + bi graphed in the complex plane is comparable to the point (a,b) graphed in the standard (x,y) coordinate plane. The modulus of the complex number a + bi is given by . Which of the complex numbers z1, z2, z3, z4, and z5 below has the greatest modulus?

Az1Bz2Cz3Dz4Ez5

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the real numbers, what is the solution of the equation 82x + 1 = 41 − x ?

Source page for act-april-2007-form-64e-math-049

A−B−C−D0EChoice E — see source image

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Paper 71 · ACT April 2007 · Form 64E · Math Test

The graph of the trigonometric function y = 2 cos冢 x冣 is shown below. The function is:

Aeven (that is, f(x) = f(−x) for all x).Bodd (that is, f(−x) = −f(x) for all x).Cneither even nor odd.Dthe inverse of a cotangent function.Eundefined at x = π.

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Paper 71 · ACT April 2007 · Form 64E · Math Test

An integer from 100 through 999, inclusive, is to be chosen at random. What is the probability that the number chosen will have 0 as at least 1 digit?

Source page for act-april-2007-form-64e-math-051

AChoice A — see source imageBChoice B — see source imageCChoice C — see source imageDChoice D — see source imageEChoice E — see source image

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the figure below, line q in the standard (x,y) coordinate plane has equation −2x + y = 1 and intersects line r, which is distinct from line q, at a point on the x-axis. The angles, ∠a and ∠b, formed by these lines and the x-axis are congruent. What is the slope of line r ?

Source page for act-april-2007-form-64e-math-052

A−2B−CChoice H — see source imageD2ECannot be determined from the given information

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the right triangle below, 0 < b < a. One of the angle measures in the triangle is tan−1 冢冣. What is cos 冤tan−1 冢冣冥? 公僓僓僓僓僓僓 a2 + b2 _____ _____ a b _____ ____ _ _ 公僓僓僓僓僓僓 a2 + b2 a _____ ______ 公僓僓僓僓僓僓 a2 + b2 b _ _ a a _ _ b 公僓僓僓僓僓僓 a2 + b2 b a a _ _ b a _ _ b 1 _ _ 2 1 _ _ 2 3 2 1 –1 O –2 –1 1 x y ∠a q r ∠b 271 _ ______ 1,000 171 _ ___ 900 90 _ ___ 900 81 _ ___ 900 19 _ ___ 900 2 –4 O –2 –6 4 6 –2 2 x y 1 _ _ 2 1 _ _ 7 1 _ _ 8 1 _ _ 4 1 _ _ 3 imaginary axis z1 z5 z4 z3 z2 real axis O 公僓僓僓僓僓僓 a2 + b2 GO ON TO THE NEXT PAGE. 2 2 ACT-64E-PRACTICE 32 The radio signal from the transmitter site of radio station WGGW can be received only within a radius of 52 miles in all directions from the transmitter site. A map of the region of coverage of the radio signal is shown below in the standard (x,y) coordinate plane, with the transmitter site at the origin and 1 coordinate unit representing 1 mile.

Source page for act-april-2007-form-64e-math-053

AChoice A — see source imageBChoice B — see source imageCChoice C — see source imageDChoice D — see source imageEChoice E — see source image

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Which of the following is closest to the area, in square miles, of the region of coverage of the radio signal?

A2,120B2,700C4,250D8,500E16,990

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Paper 71 · ACT April 2007 · Form 64E · Math Test

Which of the following is an equation of the circle shown on the map? = 52 = 522 + y2 = 52 + y2 = 522

Ax + y = 52B(x + y)2C(x + y)2Dx2Ex2

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Paper 71 · ACT April 2007 · Form 64E · Math Test

The transmitter site of radio station WGGW and the transmitter site of another radio station, WGWB, are on the same highway 100 miles apart. The radio signal from the transmitter site of WGWB can be received only within a radius of 60 miles in all directions from the WGWB transmitter site. For how many miles along the highway can the radio signals of both stations be received? (Note: Assume the highway is straight.)

A8B12C40D44E48

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Paper 71 · ACT April 2007 · Form 64E · Math Test

The graphs of the equations y = x − 1 and y = (x − 1)4 are shown in the standard (x,y) coordinate plane below. What real values of x, if any, satisfy the inequality (x − 1)4 < (x − 1) ?

ANo real valuesBx < 0 and x > 1Cx < 1 and x > 2D0 < x < 1E1 < x < 2

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Paper 71 · ACT April 2007 · Form 64E · Math Test

For every positive 2-digit number, x, with tens digit t and units digit u, let y be the 2-digit number formed by reversing the digits of x. Which of the following expressions is equivalent to x − y ?

A9(t − u)B9(u − t)C9t − uD9u − tE0

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Paper 71 · ACT April 2007 · Form 64E · Math Test

In the figure below, the vertices of 䉭ABC have (x,y) coordinates (4,5), (5,3), and (1,3), respectively. What is the area of 䉭ABC ?

A4B4公僓 2C4公僓 3D8E8公僓 2

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Paper 71 · ACT April 2007 · Form 64E · Math Test

The sum of an infinite geometric series with first term a and common ratio r < 1 is given by . The sum of a given infinite geometric series is 200, and the common ratio is 0.15. What is the second term of this series? a _____ 1 − r O x y C(1,3) A(4,5) B(5,3) y 2 1 –1 –2 O x 2 –1 1 y = (x – 1)4 y = x – 1 O y x 52 52 N S E W 2 2 ACT-64E-PRACTICE END OF TEST 2 STOP! DO NOT TURN THE PAGE UNTIL TOLD TO DO SO. DO NOT RETURN TO THE PREVIOUS TEST. 33 Use the following information to answer questions 54–56.

A25.5B30C169.85D170E199.85

Form 64E · Math Test · 60 questions

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