AP Calculus AB Related Rates Test with Point Values (PDF)
This related rates test for AP Calculus AB covers related rates (Common Core AP CALC AB 4.4), with a free PDF to print.
This test checks related rates problems with circles, spheres, cones, ladders, shadows and right triangles (AP Calculus AB, 4.4). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.
Time and points
- Grade: AP Calculus AB
- Chapter: Related Rates
- Standards: AP CALC AB 4.4
- Time: 45 minutes
- Total: 20 points
Related rates test: questions
Question 1 – Circle area [2 pts]
The radius of a circular puddle grows at 0.5 cm/s. How fast is the area growing when \(r=12\) cm?
Question 2 – Ladder [3 pts]
A 10-ft ladder leans against a wall. The top slides down at 1.5 ft/s. How fast is the base moving away from the wall when the base is 6 ft from the wall?
Question 3 – Cone volume [3 pts]
Water drains from a cone-shaped tank (point down) with radius 4 ft and height 8 ft at 6 ft³/min. How fast is the water level falling when the depth is 4 ft?
Question 4 – Rectangle [2 pts]
A rectangle’s length increases at 4 cm/s and its width decreases at 1 cm/s. At the moment when the length is 10 cm and the width is 6 cm, how fast is the area changing?
Question 5 – Shadow [3 pts]
A 5-ft person walks away from a 20-ft lamppost at 6 ft/s. How fast is the length of the shadow increasing?
Question 6 – Distance between boats [2 pts]
Boat A sails east at 12 mph and boat B sails south at 5 mph from the same dock. How fast is the distance between them growing after 1 hour?
Question 7 – Angle change [2 pts]
A runner moves along a straight track 50 m from a camera at 8 m/s. Let \(\theta\) be the angle between the camera’s line to the runner and the line perpendicular to the track. Find \(\frac{d\theta}{dt}\) when the runner is at the point 50 m along the track (so \(\theta=\frac{\pi}{4}\)).
Question 8 – Sphere and cylinder [3 pts]
A spherical balloon’s radius grows at 0.2 cm/s. (a) How fast is its volume growing when \(r=10\)? (1 point) (b) How fast is its surface area \(S=4\pi r^{2}\) growing at that moment? (1 point) (c) Which is larger numerically, and why might that be? (1 point)
Key facts behind this related rates test
- 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
Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.
All AP Calculus AB chapters: AP Calculus AB math.
