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ACT Form 74C Math Answer Explanations

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Explain to me: 



1. The top surface of a rectangular table has an area of 100 square feet and a width of 5 feet. What is the length, in feet, of the surface?

$$Length = \frac{100}{5}=20 \ feet$$

Choice C.

1. The top surface of a rectangular table has an area of 100 square feet and a width of 5 feet. What is the length, in feet, of the surface?

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2. A wallet containing 2 five-dollar bills, 9 ten-dollar bills, and 5 twenty-dollar bills is found and returned to its owner. The wallet's owner will reward the finder with 1 bill drawn randomly from the wallet. What is the probability that the bill drawn will be a twenty-dollar bill?

$$p = \frac{5}{2+9+5}=\frac{5}{16}$$

Choice J.

2. A wallet containing 2 five-dollar bills, 9 ten-dollar bills, and 5 twenty-dollar bills is found and returned to its owner. The wallet’s owner will reward the finder with 1 bill drawn randomly from the wallet. What is the probability that the bill drawn will be a twenty-dollar bill?

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3. In his costume supplies, Elmo the clown has 4 noses, 3 pairs of lips, and 2 wigs. A clown costume consists of 1 nose, 1 pair of lips, and 1 wig. How many different clown costumes can Elmo make?

$$Number \ of \ arrangement = 4\times3\times2=24$$

Choice E.

3. In his costume supplies, Elmo the clown has 4 noses, 3 pairs of lips, and 2 wigs. A clown costume consists of 1 nose, 1 pair of lips, and 1 wig. How many different clown costumes can Elmo make?

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4. Esteban and his family are making care packages to send to children at summer camp. Each complete care package contains 5 pens, 2 notebooks, 3 envelopes, 12 cookies, and 5 candy bars. Esteban and his family have already made 7 complete care packages and the following materials remain:

30 pens still remain, enough for 6 packages;

20 notebooks still remain, enough for 10 packages;

24 envelopes still remain, enough for 8 packages;

84 cookies still remain, enough for 7 packages;

45 candy bars still remain, enough for 9 packages;

Apparently, pens are the bottleneck.

Choice F.

4. Esteban and his family are making care packages to send to children at summer camp. Each complete care package contains 5 pens, 2 notebooks, 3 envelopes, 12 cookies, and 5 candy bars. Esteban and his family have already made 7 complete care packages and the following materials remain:

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5. A formula for the volume of a right circular cone is

$$V=\frac{1}{3}\cdot \frac{22}{7}\cdot 6^2\times 28=1056$$

Choice C.

5. A formula for the volume of a right circular cone is

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6. In AACD below, B is on AC, E is on AD, the measure of LCAD is 28°, and AD is perpendicular to both BE and CD. What is the measure of ZCBE ?

6. In AACD below, B is on AC, E is on AD, the measure of LCAD is 28°, and AD is perpendicular to both BE and CD. What is the measure of ZCBE ?

Choice G.

6. In AACD below, B is on AC, E is on AD, the measure of LCAD is 28°, and AD is perpendicular to both BE and CD. What is the measure of ZCBE ?

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7. What is the sum of 0.1x2 + 3x + 80 and 0.5x2 — 2x + 60 for all x ?

$$(0.1x^2+3x+80) +(0.5x^2-2x+60)=(0.1x^2+0.5x^2)+(3x-2x)+(80+60)=0.6x^2+x+140$$

Choice B.

7. What is the sum of 0.1×2 + 3x + 80 and 0.5×2 — 2x + 60 for all x ?

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8. Students studying motion observed a cart rolling at a constant rate along a straight line. The table below gives the distance, d feet, the cart was from a reference point at 1-second intervals from t = 0 seconds to t = 5 seconds.

Only d = 3t + 15 fulfils all the data in the table.

Choice H.

8. Students studying motion observed a cart rolling at a constant rate along a straight line. The table below gives the distance, d feet, the cart was from a reference point at 1-second intervals from t = 0 seconds to t = 5 seconds.

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9. Dmitry bought a pair of pants at the discounted price of $30. The original price of the pants was $40. What was the percent of the discount?

$$$40\times(1-d)=$30 \\ \Rightarrow d=25\%$$

Choice C.

9. Dmitry bought a pair of pants at the discounted price of $30. The original price of the pants was $40. What was the percent of the discount?

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10. What is the value of 1-61 — 17 — 411 ?

$$|-6|-|7-41|=6-34=-28$$

Choice G.

10. What is the value of 1-61 — 17 — 411 ?

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11. Samantha, Nyla, and Jerry own shares of stock in the Triumph Hotels company. The shares of stock that they own have a combined value of $6,880. Samantha owns 70 shares, Nyla owns 50 shares, and Jerry owns 40 shares. What is the value of the shares Samantha owns?

$$Value=$6,880\times \frac{70}{70+50+40}=$3,010$$

Choice C.

11. Samantha, Nyla, and Jerry own shares of stock in the Triumph Hotels company. The shares of stock that they own have a combined value of $6,880. Samantha owns 70 shares, Nyla owns 50 shares, and Jerry owns 40 shares. What is the value of the shares Samantha owns?

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12. A new club wants to attract customers who are at least 18 but less than 30 years of age. One of the number lines below illustrates the range of ages, in years, of the customers the club wants to attract. Which number line is it?

Only the number line in G fulfils:

$$18\leq age <30$$

Choice G.

12. A new club wants to attract customers who are at least 18 but less than 30 years of age. One of the number lines below illustrates the range of ages, in years, of the customers the club wants to attract. Which number line is it?

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13. In the figure shown below, E and G lie on AC, D and F lie on AB, DE and FG are parallel to BC, and the given lengths are in feet. What is the length of AC, in feet?

$$\frac{AC}{AB}=\frac{AE}{AD} \\ \Rightarrow \frac{AC}{6+5+4}=\frac{12}{6} \\ \Rightarrow AC=30$$

Choice D.

13. In the figure shown below, E and G lie on AC, D and F lie on AB, DE and FG are parallel to BC, and the given lengths are in feet. What is the length of AC, in feet?

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14. Which of the following integers is closest to431

$$\frac{\sqrt{50}}{2}\approx 3.54 \approx 4$$

Choice G.

14. Which of the following integers is closest to431

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15. The ratio of Jane's age to her daughter's age is 9:2. The sum of their ages is 44. How old is Jane?

$$9k+2k=44 \\ \Rightarrow k=4 \\ 9k=36$$

Choice D.

15. The ratio of Jane’s age to her daughter’s age is 9:2. The sum of their ages is 44. How old is Jane?

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16. For the next school year, a college will use	of the

$$1-\frac{1}{9}-\frac{1}{6}=\frac{13}{18}$$

Choice H.

16. For the next school year, a college will use of the

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17. What value of x makes the proportion below true?

$$35\times(10+x)=42\times10 \\ \Rightarrow 10+x=12 \\ \Rightarrow x=2$$

Choice A.

17. What value of x makes the proportion below true?

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18. The rectangle shown• in the figure below is partitioned into 3 triangles, 2 of which are shaded. What is the total area, in square inches, of the 2 shaded regions?

$$Area \ of \ 2 \ shaded \ regions = 8\times 10 – \frac{8\times10}{2}=40$$

Choice J.

18. The rectangle shown• in the figure below is partitioned into 3 triangles, 2 of which are shaded. What is the total area, in square inches, of the 2 shaded regions?

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19. Which of the following ordered pairs in the standard (x,y) coordinate plane satisfies the system of inequalities below?

Only C fulfils all three inequations.

Choice C.

19. Which of the following ordered pairs in the standard (x,y) coordinate plane satisfies the system of inequalities below?

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20. The graph • of y = 3 — 5 sin(x — n) is shown in the standard (x ,y) coordinate plane below. What is the range of y ?

The range of y = sin (x-π) is :

$$-1\leq y \leq 1$$

The range of y = -5sin (x-π) is :

$$-5\leq y \leq 5$$

The range of y = 3-5sin (x-π) is :

$$-2\leq y \leq 8$$

Choice H.

20. The graph • of y = 3 — 5 sin(x — n) is shown in the standard (x ,y) coordinate plane below. What is the range of y ?

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21. Given functions f (x) = 2x + 1 and g(x) = x2 — 4, what is

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21. Given functions f (x) = 2x + 1 and g(x) = x2 — 4, what is

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22. A fabric store sells flannel and calico fabrics. Joan pays $25 for 3 yards of flannel and 4 yards of calico. Chris pays $11 for 1 yard of flannel and 2 yards of calico. What is the price of 1 yard of calico?

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22. A fabric store sells flannel and calico fabrics. Joan pays $25 for 3 yards of flannel and 4 yards of calico. Chris pays $11 for 1 yard of flannel and 2 yards of calico. What is the price of 1 yard of calico?

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23. The scores given below were earned by 10 students on a recent biology test. What is the median score?

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23. The scores given below were earned by 10 students on a recent biology test. What is the median score?

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24. A parallelogram has a perimeter of 84 inches, and 1 of its sides measures 16 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

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24. A parallelogram has a perimeter of 84 inches, and 1 of its sides measures 16 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

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25. In the figure below, all of the small squares are equal	DO YOUR FIGURING HERE.

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25. In the figure below, all of the small squares are equal DO YOUR FIGURING HERE.

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26. A bag contains 16 red'marbles, 7 yellow marbles, and 19 green marbles. How many additional red marbles must be added to the 42 marbles already in the bag so that the probability of randomly drawing a red marble

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26. A bag contains 16 red’marbles, 7 yellow marbles, and 19 green marbles. How many additional red marbles must be added to the 42 marbles already in the bag so that the probability of randomly drawing a red marble

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27. For all a > 0, which of the following expressions is equal to a-2 ?” width=”300″></a></p>
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27. For all a > 0, which of the following expressions is equal to a-2 ?

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28. Jamie claims,

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28. Jamie claims, “If a triangle is in Set A, then it is not isosceles.” Later, Jamie discovers that AMNP is a counterexample proving this claim false. Which of the following statements must be true about AMNP ?

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29. What is the area, in square coordinate units, of ABCD ?

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29. What is the area, in square coordinate units, of ABCD ?

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30. What is the distance, in coordinate units, from B to D ?

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30. What is the distance, in coordinate units, from B to D ?

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31. What is the slope of BC ?

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31. What is the slope of BC ?

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32. Parallelogram ABCD will be reflected over the y-axis. What will be the coordinates of the image of A ?

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32. Parallelogram ABCD will be reflected over the y-axis. What will be the coordinates of the image of A ?

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33. Which of the following is equivalent to 82 - 40.5 ?

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33. Which of the following is equivalent to 82 – 40.5 ?

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34. A school admissions office accepts 2 out of every 7 applicants. Given that the school accepted 630 students, how many applicants were NOT accepted?

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34. A school admissions office accepts 2 out of every 7 applicants. Given that the school accepted 630 students, how many applicants were NOT accepted?

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35. What is the value of log2 1,/g ?

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35. What is the value of log2 1,/g ?

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36. Jie asked 90 students to choose 1 favorite fruit from

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36. Jie asked 90 students to choose 1 favorite fruit from

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37. For all real numbers x such that x # 0, 4 7.; + 7 — = ?

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37. For all real numbers x such that x # 0, 4 7.; + 7 — = ?

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38. What fraction of these nonstudent residents replied that they disapproved of the change in hours?

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38. What fraction of these nonstudent residents replied that they disapproved of the change in hours?

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39. Suppose a person will be chosen at random from these 560 residents. Which of the following values is closest to the probability that the person chosen will NOT be a high school student and will NOT have replied with no opinion?

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39. Suppose a person will be chosen at random from these 560 residents. Which of the following values is closest to the probability that the person chosen will NOT be a high school student and will NOT have replied with no opinion?

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40. After constructing the table, it was discovered that the student status of 15 residents who replied that they approved had been incorrectly classified as nonstudents. After correcting the errors, exactly 60% of the college students had replied that they approved. To the nearest 1%, what percent of high school students replied that they approved?

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40. After constructing the table, it was discovered that the student status of 15 residents who replied that they approved had been incorrectly classified as nonstudents. After correcting the errors, exactly 60% of the college students had replied that they approved. To the nearest 1%, what percent of high school students replied that they approved?

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41. Set A and Set B each consist of 5 distinct numbers. The 2 sets contain identical numbers with the exception of the number with the least value in each set. The number with the least value in Set B is greater than the number with the least value in Set A. The value of which of the following measures must be greater for Set B than- for Set A?

A.

To simplify matters, let’s make the least value in set A 2, set B 3:

From the least to the largest, the two sets are:

Set A:

{2, x, y, z, m}

Set B:

{3, x, y, z, m}

Therefore the mean of set A = (2 + x +y +z+m)÷5 ‹ set B = (3+ x +y +z+m)÷5.

The medians for both sets are the same (i.e. y).

The modes may be the same, and may be different.

41. Set A and Set B each consist of 5 distinct numbers. The 2 sets contain identical numbers with the exception of the number with the least value in each set. The number with the least value in Set B is greater than the number with the least value in Set A. The value of which of the following measures must be greater for Set B than- for Set A?

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42. For all x such that 0 x < 90, which of the following . expressions is NOT equal to sin x° ?

G.

sin(-x°) = -sin x°

42. For all x such that 0 x < 90, which of the following . expressions is NOT equal to sin x° ?

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43. A 3-inch-tall rectangular box with a square base is constructed to hold a circular pie that has a diameter of

D.

Volume of box = 8 × 8 × 3 = 192 cubic inches

43. A 3-inch-tall rectangular box with a square base is constructed to hold a circular pie that has a diameter of

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44. Quadrilateral ABCD is shown in the figure below with the lengths of the 4 sides given in meters. The measure of ZC is 90°. What is tan A ?

G.

Since ΔBCD is a right triangle, BD = 5, which means ΔBDA is a right triangle too (5² + 12² = 13²), therefore, tan A = BD/AD = 5/12.

44. Quadrilateral ABCD is shown in the figure below with the lengths of the 4 sides given in meters. The measure of ZC is 90°. What is tan A ?

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45. Given today is Tuesday, what day of the week was it 200 days ago?

D.

200 days ago is equivalent to 200 ÷ 7 = 28 weeks and 4 days ago

28 weeks ago, it was still a Tuesday. Counting backwards for 4 days, it would be Monday, Sunday, Saturday and Friday. Therefore it was Friday.

45. Given today is Tuesday, what day of the week was it 200 days ago?

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46. In the figure below, line m is perpendicular to line n, and both lines intersect line q at the same point. The measure of L1 is (3x — 10)°, and the measure of L2 is (2x + 10)°. What is the measure of L3 ?

G.

∠1 = ∠2
→3x – 10 = 2x + 10
→x = 20
→∠1 =3×20-10=50°
→∠3 = 180° – ∠1 – 90°=40°

46. In the figure below, line m is perpendicular to line n, and both lines intersect line q at the same point. The measure of L1 is (3x — 10)°, and the measure of L2 is (2x + 10)°. What is the measure of L3 ?

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47. The greatest common factor of 2 whole numbers is 10. The least common multiple of these same 2 numbers is 120. What are the 2 numbers?

E.

The best way to solve this type of questions is to try each option (start from the last one! From my experience, the correct answer is usually the last one or almost the last.)

47. The greatest common factor of 2 whole numbers is 10. The least common multiple of these same 2 numbers is 120. What are the 2 numbers?

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48. The side lengths of a certain triangle are 4, 5, and 7 centimeters. Which of the following descriptions best classifies this triangle?

H.

The Scalene Triangle has no congruent sides. In other words, each side must have a different length.

An obtuse triangle is a triangle in which one of the angles is an obtuse angle. (Obviously, only a single angle in a triangle can be obtuse or it wouldn’t be a triangle.) A triangle must be either obtuse, acute, or right.

The triangle in question has all sides different in size, and thus is a Scalene triangle.

For a right triangle to have its two smaller sides (legs) to be 4 and 5, its largest side (hypotenuse) must be √(4²+5²) = 6.4. And in the question the largest side is 7 which is larger than 6.4, making the angle larger than 90°, hence an obtuse angle, the triangle an obtuse triangle.

48. The side lengths of a certain triangle are 4, 5, and 7 centimeters. Which of the following descriptions best classifies this triangle?

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49. A professional baseball team will play 1 game Saturday • and 1 game Sunday. A sportswriter estimates the team has a 60% chance of winning on Saturday but only a 35% chance of winning on Sunday. Using the sportswriter's estimates, what is the probability that the team will lose both games?

D.

A sportswriter estimates the team has a 60 % chance of winning on Saturday but only a 35% chance of winning on Sunday.

→ The team has a 40% chance of losing on Saturday and a 65% chance of losing on Sunday.

And the two events are independent. Therefore the probability that the team loses both games are 40% × 65% = 26%

49. A professional baseball team will play 1 game Saturday • and 1 game Sunday. A sportswriter estimates the team has a 60% chance of winning on Saturday but only a 35% chance of winning on Sunday. Using the sportswriter’s estimates, what is the probability that the team will lose both games?

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50. The graph of f(x)	2	X- 3 is shown below. What is

J.

The denominator of f(x) = (x-3)(x+1) ≠ 0

Therefore x ≠ 3 / -1

50. The graph of f(x) 2 X- 3 is shown below. What is

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51. Get-A-Great-Read Books is adding a new phone line. The phone company says that the first 3 digits of the phone number must be 555, but the remaining 4 digits, where each digit is a digit from 0 through 9, can be chosen by Get-A-Great-Read Books. How many phone numbers are possible?

E.

555_ _ _ _

Each digit has 10 possibilities (from 0 through 9), therefore the total number of possible numbers are 10 × 10 × 10 × 10 = 10^4

51. Get-A-Great-Read Books is adding a new phone line. The phone company says that the first 3 digits of the phone number must be 555, but the remaining 4 digits, where each digit is a digit from 0 through 9, can be chosen by Get-A-Great-Read Books. How many phone numbers are possible?

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52. In the standard (x,y) coordinate plane, the circle centered at (1,3) that passes through (4,7) is the set of all points that are:

F.

The circle represents all dots that are 5 units from A.

52. In the standard (x,y) coordinate plane, the circle centered at (1,3) that passes through (4,7) is the set of all points that are:

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53. Which of the following values is the x-coordinate of the point in the standard (x,y) coordinate plane where the graph of the line y = 7 intersects the graph of the function y = ln(x — 2) + 3 ?

B.

The question asks what x is when y = 7.

7 = ln(x-2) + 3

ln(x-2)=4=ln (e^4)

→x-2=e^4

→x = e^4 + 2

53. Which of the following values is the x-coordinate of the point in the standard (x,y) coordinate plane where the graph of the line y = 7 intersects the graph of the function y = ln(x — 2) + 3 ?

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54. Three copy machines—A, B, and C—copy at the same rate and will all be used to make copies of a report. At 8:00 a.m., all 3 machines begin copying. Machine A breaks down at 10:00 a.m. and is back in service at 1:00 p.m. Machine B breaks down at 12:00 p.m. (noon) and begins copying again at 3:00 p.m. All 3 machines finish copying at 5:00 p.m. when the copying of the report is complete. One of the following graphs shows n, the number of copies made, as a function of t, the time at any given point during the copying. Which graph is it?	DO YOUR FIGURING HERE.

F.

The slope represents the copying rate. The steeper the line is, the higher the copying rate.

Machine A breaks down at 10:00 a.m. and is back in service at 1:00 p.m. Machine B breaks down at 12:00 p.m. (noon) and begins copying again at 3:00 p.m.

→ The line must go flatter at 10 am. And during 12-3pm, the line must go even flatter. Only F fulfils this.

54. Three copy machines—A, B, and C—copy at the same rate and will all be used to make copies of a report. At 8:00 a.m., all 3 machines begin copying. Machine A breaks down at 10:00 a.m. and is back in service at 1:00 p.m. Machine B breaks down at 12:00 p.m. (noon) and begins copying again at 3:00 p.m. All 3 machines finish copying at 5:00 p.m. when the copying of the report is complete. One of the following graphs shows n, the number of copies made, as a function of t, the time at any given point during the copying. Which graph is it? DO YOUR FIGURING HERE.

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55. A sporting-goods store sells baseball caps for $22 each. At this price, 40 caps are sold per week. For every $1 decrease in price, the store will sell 4 more caps per week. The store will adjust the price to maximize revenue. What will be the maximum possible revenue for 1 week?

E.

Let r be revenue, n the number of caps, p the price per cap
r = p·n

There is also a linear relationship between n and p:
n=k·p+a

A sporting-goods store sells baseball caps for $22 each. At this price, 40 caps are sold per week.

→ 40 = 22·k + a

For every $1 decrease in price, the store will sell 4 more caps per week.

→ k=-4
⇒ a=128
⇒n = -4p + 128
⇒r = -4p² + 128p = -4(p-16)² + 1024

r has a maximum value of 1024.

55. A sporting-goods store sells baseball caps for $22 each. At this price, 40 caps are sold per week. For every $1 decrease in price, the store will sell 4 more caps per week. The store will adjust the price to maximize revenue. What will be the maximum possible revenue for 1 week?

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56. Each of the following graphs in the standard (x,5)	DO YOUR FIGURING HERE.

K.

The quickest way would be plugging in numbers for a, b, and c that satisfy the condition. Let a=1, b=2, c=3, we have:

x + 2y ≤ 3

where the function x + 2y = 3 crosses (0,1.5) and (3,0).

Because (0,0) satisfies x + 2y ≤ 3, the side which includes (0,0) should be the area that represents the x + 2y ≤ 3.

56. Each of the following graphs in the standard (x,5) DO YOUR FIGURING HERE.

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57. The art club designed and made banners of the school colors, blue and white, for their fund-raiser. Each

A.

To make one banner, ¼ yard of blue material and 3/8 yard of white material are needed.

Thus, to make 500 banners, 125 (¼ × 500) yards of blue material and 187.5(3/8 × 500)  yards of white material are needed.

To buy in bolts and to have at least 125 yards of blue material, 13 (125÷10=12.5) bolts of blue material must be bought, leaving 5 ((13-12.5)×10=5) yards of extra blue material, which is enough for 20 (5÷¼=20) banners.

To buy in bolts and to have at least 187.5 yards of white material, 16 (187.5÷12=15.625) bolts of white material must be bought, leaving 4.5 [(16-15.625)×12=4.5) yards of extra white material, which is enough for 12 (4.5÷3/8=12) banners.

With the extra white material being the bottleneck, in the end 12 extra banners can be made.

57. The art club designed and made banners of the school colors, blue and white, for their fund-raiser. Each

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58. For all real numbers x and the imaginary number i, which of the following expressions is equivalent to

F.

What you need to know:

i² = -1

(x-3i)³
= (x-3i)·[x²-6xi+(3i)²] = (x-3i)·[x²-6xi+9i²]
= (x-3i)·[x²-6xi+9·(-1)]
= (x-3i)·[x²-6xi-9]
=x³-6·x²·i-9x-3x²·i+18·x·i²+27i
= x³-9x²i-27x+27i

58. For all real numbers x and the imaginary number i, which of the following expressions is equivalent to

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59. The graph in the standard (x,y) coordinate plane below is the graph of one of the following functions. Which one?

D.

According to the graph, the function equals to 0 when x is -4, 0, and 6. And of the five functions, only p(x) fulfils this requirement.

59. The graph in the standard (x,y) coordinate plane below is the graph of one of the following functions. Which one?

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60. The table below shows the numbers of rows and columns in each of 5 matrices.

K.

Requirements for Matrix Multiplication:

In order to multiply two matrices, A and B, the number of columns in A must equal the number of rows in B.

That is, # of columns in A = # of rows in B

Since # of columns in A (= n) ≠ # of rows in C (= k), AC is not possible.

60. The table below shows the numbers of rows and columns in each of 5 matrices.

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