## ACT Prep 3 Answers

- Question #1: What is sec(a)sin(a) + csc(a)cos(a) , if a = 45
^{o}?

(a) √2

(b) 2

(c) 1

(d) 1/√2

(e) 1/2Answer: sec(a)sin(a) + csc(a)cos(a) = [1/cos(a)]sin(a) + [1/sin(a)]cos(a) = sin(a)/cos(a) + cos(a)/sin(a) = (√2/2)/(√2/2) + (√2/2)/(√2/2) = 1 + 1 = 2.

- Question #2: In the standard (x,y) coordinate plane, 3 corners of a square are (4,–3), (–4,3), and (–4,-3). What are the coordinates of the square’s fourth corner?

(a) (1/4, -1/3)

(b) (4, 3)

(c) (3, 4)

(d) (3, -4)

(e) (3, -4)Answer:

The easiest way to solve this problem is to sketch the 3 corners of the square. We notice that the fourth corner of the square is (4, 3). - Question #3: Prizes totaling $60,000 were awarded unequally between 3 contestants. Which of the following choices could be the highest prize?(a) $5,000

(b) $10,000

(c) $15,000

(d) $25,000

(e) $65,000Answer: The highest prize should be higher than one third of $60,000 and lower than $60,000. The only answer that satisfies this condition is $25,000.

- Question #4: What is the area of rectangle ABCD if the length of the side AB is 5?

(a) 40

(b) 50

(c) 25

(d) 25√3

(e) 15√3Answer: tan(CAD) = CD/AD

1/√3 = 5/AD

AD = 5√3

The area of rectangle ABCD is AD·BC = 5√3·5 = 25√3. - Question #5: What is the circumference of circle O from the figure below if the length of the sides of the ABCD square is 5 inches?(a) 10·¶ inches

(b) 5·¶ inches

(c) ¶ inches

(d) 16 inches

(e) 12; inchesAnswer: Angle BOA is 60

^{o}and OA = OB, so triangle BOA is equilateral, and the radius of the circle is equal to the side of the square ABCD.

The circumference of the circle is 2·¶5 = 10·¶ - Question #6: Ellipse (x – 3)
^{2}/9 + (y – 5)^{2}/5 = 1 is inscribed in the circle O. The transverse of the ellipse is equal to the diameter of the circle O. The equation that describes circle O in the standard (x,y) plane is:

(a) (x + 3)^{2}+ (y + 5)^{2}= 3^{2}

(b) (x – 3)^{2}+ (y – 5)^{2}= 1

(c) (x – 3)^{2}+ (y – 5)^{2}= 3^{2}

(d) (x – 3)^{3}+ (y – 5)^{3}= 3^{3}

(e) (x – 3)^{2}+ (y – 5)^{2}= 7Answer: If a>b, the transverse of the ellipse (x – m)

^{2}/a^{2}+ (y – n)^{2}/b^{2}= 1 is equal to 2a.

In our case the transverse diameter that is equal to the diameter of the circle is 2·3 = 6, so the radius of the circle is 3.

The center of the circle is (3, 5) in the (x, y) plane.

A circle with a center in (3, 5) and of radius 3, will be described by the equation:

(x – 3)^{2}+ (y – 5)^{2}= 3^{2} - Question #7: The marked price of shirts in a store is $25. During one week, the store sells m shirts at the marked price and n shirts discounted 25% of the marked price. Which of the following is an expression of the average price of the shirts sold?(a) ($25m + $18.75n) /(m + n)

(b) ($25m + $18.75n) /(2m + 2n)

(c) $25m + $18.75n

(d) ($25 + $20) /(m + n)

(e) (m + n) / ($25m + $20n)Answer: The price of a discounted shirt is (1 – .25)$25 = $18.75.

The total revenue during the week is ($25m + $18.75n) and the number of shirts sold is (m + n).

In conclusion the average price of the shirts is ($25m + $18.75n)/(m + n). - Question #8: A box contains 7 blue marbles, 4 red marbles and 6 green marbles. How many additional blue marbles must be added in the box so that the probability of randomly drawing a blue marble is 1/2?
(a) 2

(b) 3

(c) 4

(d) 5

(e) 6Answer: If the probability of randomly drawing a blue marble is 1/2 after x blue marbles were added, the number of blue marbles must be half of the total numbers of marbles.

1/2 = (x + 7)/(x + 7 + 4 + 6)

x = 3 - Question #9: A number is increased 50% and the resulting number is decreased 50%. The final number is what percent of the original number?(a) 50%

(b) 125%

(c) 75%

(d) 100%

(e) 105%Answer: If the original number is x, after it is increased 50% the resulting number is 1.5x.

1.5x is decreased 50% so the final number is .5·1.5 = .75·x or 75% of x. - Question #10: The inequality 2x + 3 > 5x – 6 is equivalent with which of the following inequalities?(a) x > 6

(b) x < 6

(c) x > 3

(d) x < 3

(e) x < 10Answer:2x + 3 > 5x – 6

9 > 3x

3 > x

x < 3

Examples of ACT geometry questions

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