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AP Calculus BC Practice Quiz: Solving Optimization Problems

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

A company's profit, P, in thousands of dollars, from selling x hundred items is modeled by the function P(x). An analysis shows that the maximum value of P(x) is 75. What is the correct interpretation of this value?

All Questions (7)

A company's profit, P, in thousands of dollars, from selling x hundred items is modeled by the function P(x). An analysis shows that the maximum value of P(x) is 75. What is the correct interpretation of this value?

A) The company must sell 75 items to maximize its profit.

B) The maximum possible profit the company can make is $75,000.

C) The price of each item sold is $75.

D) The cost to produce the items that yield maximum profit is $75,000.

Correct Answer: B

The function P(x) represents the profit in thousands of dollars. Therefore, the maximum value of the function P(x) corresponds to the maximum possible profit. Since P(x) is in thousands of dollars, a value of 75 means $75,000.

The area, A, in square meters, of a rectangular garden with a fixed amount of fencing is a function of its width, w, in meters. The function A(w) is found to have a maximum value of 225. What is the significance of the number 225 in this context?

A) The total length of fencing available for the garden is 225 meters.

B) The width of the garden that results in the largest area is 225 meters.

C) The largest possible area the garden can have is 225 square meters.

D) The perimeter of the largest possible garden is 225 meters.

Correct Answer: C

The function A(w) models the area of the garden. The maximum value of this function represents the largest possible output, which in this applied context is the maximum possible area the garden can enclose with the given fencing.

The height above the ground of a thrown javelin is given by the function h(t), where h is in feet and t is the time in seconds after it is released. The function has a maximum value of 25 at t = 2. What does the value 25 represent?

A) The javelin is in the air for a total of 25 seconds.

B) The initial speed of the javelin was 25 feet per second.

C) The greatest height the javelin reaches is 25 feet.

D) The time it takes for the javelin to reach its highest point is 25 seconds.

Correct Answer: C

The function h(t) represents the height of the javelin. The maximum value of this function corresponds to the peak of its trajectory. Therefore, 25 is the maximum height, in feet, that the javelin achieves.

A manufacturer determines that the cost, C, in dollars, to produce a cylindrical can of a specific volume is a function of the can's radius, r. The function C(r) has a minimum value of 1.25. What is the meaning of this value in this context?

A) The radius of the most cost-effective can is 1.25 inches.

B) The volume of the can is 1.25 cubic inches.

C) The lowest possible production cost for one can is $1.25.

D) The height of the most cost-effective can is 1.25 inches.

Correct Answer: C

The function C(r) models the production cost. A minimum value of this function represents the lowest possible cost to produce the can while meeting the volume requirement. The input variable is the radius, r, while the output is the cost, C.

The concentration of a medication in a patient's bloodstream, C, in milligrams per liter (mg/L), is modeled by C(t), where t is the time in hours after injection. The maximum value of C(t) is found to be 15. What is the correct interpretation of this value?

A) The medication is effective for a maximum of 15 hours.

B) The peak concentration of the medication in the bloodstream is 15 mg/L.

C) The time it takes for the medication to reach its peak concentration is 15 hours.

D) The total amount of medication injected was 15 milligrams.

Correct Answer: B

The function C(t) represents the concentration of the medication. The maximum value of this function is the highest possible output, which corresponds to the peak concentration level the medication reaches in the bloodstream.

The total annual cost, C, for a company to manage its inventory is modeled by the function C(x), where x is the number of items ordered in each shipment. This cost includes both ordering fees and storage costs. The function has a minimum value of $40,000 when x = 500. What is the practical meaning of the value $40,000?

A) The cost to place a single order for 500 items is $40,000.

B) The ideal number of items to order per shipment is 40,000.

C) The lowest possible total annual cost for managing inventory is $40,000.

D) The annual cost of storing 500 items in the warehouse is $40,000.

Correct Answer: C

The function C(x) represents the total annual inventory cost. Its minimum value is the lowest possible value for this total cost. Therefore, $40,000 is the most economically efficient total cost the company can achieve for its inventory per year. The value x=500 is the order size that leads to this minimum cost.

A car's fuel efficiency, E, in miles per gallon (mpg), is a function of its speed, s, in miles per hour (mph), given by E(s). The function has a maximum value of 38. Which statement provides the best interpretation of this maximum value?

A) The car's maximum speed is 38 mph.

B) The car can travel a maximum of 38 miles on a full tank of gas.

C) The highest fuel efficiency the car can achieve is 38 mpg.

D) The speed at which the car is most fuel-efficient is 38 mph.

Correct Answer: C

The function E(s) models the fuel efficiency in mpg. The maximum value of this function represents the best possible fuel efficiency the car can attain. The speed 's' that yields this efficiency is the input, while the efficiency 'E' is the output.