ℓ = 24 + 3.5m
One end of a spring is attached to a ceiling. When an object of mass m kilograms is attached to the other end of the spring, the spring stretches to a length of ℓ centimeters as shown in the equation above. What is m when ℓ is 73 ?

Solution
The value of m when ℓ is 73 can be found by substituting the 73 for ℓ in ℓ = 24 + 3.5m and then solving for m. The resulting equation is 73 = 24 + 3.5m; subtracting 24 from each side gives 49 = 3.5m. Then, dividing each side of 49 = 3.5m by 3.5 gives 14 = m. Therefore, when ℓ is 73, m is 14.
Choice B is incorrect and may result from adding 24 to 73, instead of subtracting 24 from 73, when solving 73 = 24 + 3.5m. Choice C is incorrect because 73 is the given value for ℓ, not for m. Choice D is incorrect and may result from substituting 73 for m, instead of for ℓ, in the equation ℓ = 24 + 3.5m.
A quality control manager at a factory selects 7 lightbulbs at random for inspection out of every 400 lightbulbs produced. At this rate, how many lightbulbs will be inspected if the factory produces 20,000 lightbulbs?

Solution
The quality control manager selects 7 lightbulbs at random for inspection out of every 400 lightbulbs produced. A quantity of 20,000 lightbulbs is equal to \(\frac{20,000}{400}\)= 50 batches of 400 lightbulbs. Therefore, at the rate of 7 lightbulbs per 400 lightbulbs produced, the quality control manager will inspect a total of 50 × 7 = 350 lightbulbs.
Choices A, C, and D are incorrect and may result from calculation errors or misunderstanding of the proportional relationship.
A musician has a new song available for downloading or streaming. The musician earns $0.09 each time the song is downloaded and $0.002 each time the song is streamed. Which of the following expressions represents the amount, in dollars, that the musician earns if the song is downloaded d times and streamed s times?

Solution
Since the musician earns $0.09 for each download, the musician earns 0.09d dollars when the song is downloaded d times. Similarly, since the musician earns $0.002 each time the song is streamed, the musician earns 0.002s dollars when the song is streamed s times. Therefore, the musician earns a total of 0.09d + 0.002s dollars when the song is downloaded d times and streamed s times.
Choice A is incorrect because the earnings for each download and the earnings for time streamed are interchanged in the expression. Choices B and D are incorrect because in both answer choices, the musician will lose money when a song is either downloaded or streamed. However, the musician only earns money, not loses money, when the song is downloaded or streamed.
refer to the following information.
If shoppers enter a store at an average rate of r shoppers per minute and each stays in the store for an average time of T minutes, the average number of shoppers in the store, N, at any one time is given by the formula N rT = .This relationship is known as Little’s law.
The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little’s law to estimate that there are 45 shoppers in the store at any time.
The owner of the Good Deals Store opens a new store across town. For the new store, the owner estimates that, during business hours, an average of 90 shoppers per hour enter the store and each of them stays an average of 12 minutes. The average number of shoppers in the new store at any time is what percent less than the average number of shoppers in the original store at any time? (Note: Ignore the percent symbol when entering your answer. For example, if the answer is 42.1%, enter 42.1)

Solution
The correct answer is 60. The estimated average number of shoppers in the original store at any time is 45. In the new store, the manager estimates that an average of 90 shoppers per hour enter the store, which is equivalent to 1.5 shoppers per minute. The manager also estimates that each shopper stays in the store for an average of 12 minutes. Thus, by Little’s law, there are, on average, N = rt = (1.5)(12) = 18 shoppers in the new store at any time. This is \(\frac{45 − 18}{45}\) × 100 = 60 percent less than the average number of shoppers in the original store at any time.
refer to the following information.
If shoppers enter a store at an average rate of r shoppers per minute and each stays in the store for an average time of T minutes, the average number of shoppers in the store, N, at any one time is given by the formula N rT = .This relationship is known as Little’s law.
The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little’s law to estimate that there are 45 shoppers in the store at any time.
Little’s law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spend an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store?

Solution
The correct answer is 7. The average number of shoppers, N, in the checkout line at any time is N = rt, where r is the number of shoppers entering the checkout line per minute and T is the average number of minutes each shopper spends in the checkout line. Since 84 shoppers per hour make a purchase, 84 shoppers per hour enter the checkout line. This needs to be converted to the number of shoppers per minute. Since there are 60 minutes in one hour, the rate is \(\frac{84\, shoppers}{60\, minutes}\) = 1.4 shoppers per minute. Using the given formula with r = 1.4 and t = 5 yields N = rt = (1.4)(5) = 7. Therefore, the average number of shoppers, N, in the checkout line at any time during business hours is 7.
y ≤ – 15x + 3000
y ≤ 5x
In the xy‑plane, if a point with coordinates (a, b) lies in the solution set of the system of inequalities above,what is the maximum possible value of b ?

Solution
The correct answer is 750. The inequalities y ≤ −15x + 3000 and y ≤ 5x can be graphed in the xyplane. They are represented by the halfplanes below and include the boundary lines y = −15x + 3000 and y = 5x, respectively. The solution set of the system of inequalities will be the intersection of these halfplanes, including the boundary lines, and the solution (a, b) with the greatest possible value of b will be the point of intersection of the boundary lines. The intersection of boundary lines of these inequalities can be found by setting them equal to each other: 5x = −15x + 3000, which has solution x = 150. Thus, the xcoordinate of the point of intersection is 150. Therefore, the ycoordinate of the point of intersection of the boundary lines is 5(150) = −15(150) + 3000 = 750. This is the maximum possible value of b for a point (a, b) that is in the solution set of the system of inequalities.
An online store receives customer satisfaction ratings between 0 and 100, inclusive. In the first 10 ratings the store received, the average (arithmetic mean) of the ratings was 75. What is the least value the store can receive for the 11th rating and still be able to have an average of at least 85 for the first 20 ratings?

Solution
The correct answer is 50. The mean of a data set is the sum of the values in the data set divided by the number of values in the data set. The mean of 75 is obtained by finding the sum of the first 10 ratings and dividing by 10. Thus, the sum of the first 10 ratings was 750. In order for the mean of the first 20 ratings to be at least 85, the sum of the first 20 ratings must be at least (85)(20) = 1700. Therefore, the sum of the next 10 ratings must be at least 1700 − 750 = 950. The maximum rating is 100, so the maximum possible value of the sum of the 12th through 20th ratings is 9 × 100 = 900. Therefore, for the store to be able to have an average of at least 85 for the first 20 ratings, the least possible value for the 11th rating is 950 − 900 = 50.
In a circle with center O, central angle AOB has a measure of \(\frac{5\pi }{4}\) radians. The area of the sector formed by central angle AOB is what fraction of the area of the circle?

Solution
The correct answer is ^{5}⁄_{8} or .625. A complete rotation around a point is 360° or 2π radians. Since the central angle AOB has measure \(\frac{5\pi }{4}\) radians, it represents \(\frac{\frac{5\pi }{4}}{2\pi }\) = ^{5}⁄_{8} of a complete rotation around point O. Therefore, the sector formed by central angle AOB has area equal to ^{5}⁄_{8} the area of the entire circle. Either the fraction ^{5}⁄_{8} or its decimal equivalent, .625, may be gridded as the correct answer.
(−3x^{2} + 5x − 2) − 2(x^{2} − 2x − 1)
If the expression above is rewritten in the form ax^{2} + bx + c, where a, b, and c are constants, what is the value of b ?

Solution
The correct answer is 9. To rewrite the difference (−3x^{2} + 5x − 2) − 2(x^{2} − 2x − 1) in the form ax^{2} + bx + c, the expression can be simplified by using the distributive property and combining like terms as follows:
(−3x^{2} + 5x − 2) − (2x^{2} − 4x − 2)
(−3x^{2} − 2x^{2}) + (5x − (−4x)) + (−2 −(−2))
−5x^{2} + 9x + 0
The coefficient of x is the value of b, which is 9.
Alternatively, since b is the coefficient of x in the difference (−3x^{2} + 5x − 2) − 2(x^{2} − 2x − 1), one need only compute the xterm in the difference. The xterm is 5x − 2(−2x) = 5x + 4x = 9x, so the value of b is 9.
Ages of the First 12 United States Presidents at the Beginning of Their Terms in Office
The table above lists the ages of the first 12 United States presidents when they began their terms in office. According to the table, what was the mean age, in years, of these presidents at the beginning of their terms? (Round your answer to the nearest tenth.)

Solution
The correct answer is 58.6. The mean of a data set is determined by calculating the sum of the values and dividing by the number of values in the data set. The sum of the ages, in years, in the data set is 703, and the number of values in the data set is 12. Thus, the mean of the ages, in years, of the first 12 United States presidents at the beginning of their terms is \(\frac{703}{12}\). The fraction \(\frac{703}{12}\) cannot be entered into the grid, so the decimal equivalent, rounded to the nearest tenth, is the correct answer. This rounded decimal equivalent is 58.6.