AP Calculus AB Related Rates Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Related Rates – Maths PDF

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This related rates test answers for AP Calculus AB covers related rates (Common Core AP CALC AB 4.4), with a free PDF to print.

Use this answer key to score the related rates test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State Standards: AP CALC AB 4.4

Related rates test answers: worked solutions

Question 1 – Circle area [2 pts]

\(\frac{dA}{dt}=2\pi r\frac{dr}{dt}=2\pi(12)(0.5)=12\pi\).

Final answer: \(12\pi\approx 37.7\text{ cm}^{2}\text{/s}\).

Scoring: 1 point for the differentiated equation; 1 point for the value with units.

Question 2 – Ladder [3 pts]

\(x^{2}+y^{2}=100\), so \(x\frac{dx}{dt}+y\frac{dy}{dt}=0\). At \(x=6\), \(y=8\).

\(6\frac{dx}{dt}+8(-1.5)=0\) gives \(\frac{dx}{dt}=2\).

Final answer: \(2\text{ ft/s}\).

A 10-foot ladder with its base 6 feet from the wall.

Scoring: 1 point for the equation and its derivative; 1 point for the values x = 6, y = 8; 1 point for the answer with units.

Question 3 – Cone volume [3 pts]

Similar triangles: \(r=\frac{h}{2}\), so \(V=\frac{\pi}{12}h^{3}\) and \(\frac{dV}{dt}=\frac{\pi}{4}h^{2}\frac{dh}{dt}\).

\(-6=\frac{\pi}{4}(16)\frac{dh}{dt}=4\pi\frac{dh}{dt}\), so \(\frac{dh}{dt}=-\frac{3}{2\pi}\).

Final answer: \(-\frac{3}{2\pi}\approx -0.477\text{ ft/min}\).

Scoring: 1 point for r in terms of h; 1 point for the derivative; 1 point for the value.

Question 4 – Rectangle [2 pts]

\(\frac{dA}{dt}=\frac{dL}{dt}W+L\frac{dW}{dt}=4(6)+10(-1)=14\).

Final answer: \(14\text{ cm}^{2}\text{/s}\) (increasing).

Scoring: 1 point for the product rule; 1 point for the value.

Question 5 – Shadow [3 pts]

With \(x\) the distance from the post and \(s\) the shadow length: \(\frac{20}{x+s}=\frac{5}{s}\), so \(20s=5x+5s\) and \(s=\frac{x}{3}\).

\(\frac{ds}{dt}=\frac{1}{3}(6)=2\).

Final answer: \(2\text{ ft/s}\).

A 20-foot light and a 5-foot person.

Scoring: 1 point for similar triangles; 1 point for s in terms of x; 1 point for the rate.

Question 6 – Distance between boats [2 pts]

After 1 hour: \(x=12\), \(y=5\), \(D=13\). \(D\frac{dD}{dt}=x\frac{dx}{dt}+y\frac{dy}{dt}\) gives \(13\frac{dD}{dt}=12(12)+5(5)=169\).

Final answer: \(13\text{ mph}\).

Scoring: 1 point for the equation; 1 point for the value.

Question 7 – Angle change [2 pts]

\(\tan\theta=\frac{x}{50}\), so \(\sec^{2}\theta\frac{d\theta}{dt}=\frac{1}{50}\frac{dx}{dt}\). At \(\theta=\frac{\pi}{4}\): \(2\frac{d\theta}{dt}=\frac{8}{50}\).

Final answer: \(0.08\text{ rad/s}\).

Scoring: 1 point for the differentiated equation; 1 point for the value.

Question 8 – Sphere and cylinder [3 pts]

\(\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}=4\pi(100)(0.2)=80\pi\approx 251.3\text{ cm}^{3}\text{/s}\).
\(\frac{dS}{dt}=8\pi r\frac{dr}{dt}=8\pi(10)(0.2)=16\pi\approx 50.3\text{ cm}^{2}\text{/s}\).
The volume rate is larger (ratio \(\frac{r}{2}=5\)) because volume grows with the cube of the radius while area grows with the square.

A cone-shaped paper cup.

Scoring: 1 point per part.

Total score

Total

20 points

Key facts behind this related rates test answers

Key facts

Differentiate the relating equation with respect to time t, using the chain rule on every variable.
Substitute instant values only after differentiating; constants such as a ladder length stay constant.
A quantity that is decreasing has a negative rate.
Use similar triangles or a given ratio to reduce a formula to one variable before differentiating.

How to use this related rates test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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