The classic related rates scenarios — balloons, ladders, cones, shadows, converging cars — worked with implicit differentiation with respect to time. 12 problems, answer key at the bottom.
The Related Rates Method
1. Set up an equation
relate the quantities geometrically
Sphere volume, Pythagorean theorem, similar triangles — whatever connects the changing quantities at any instant.
2. Differentiate with respect to t
implicitly, using the chain rule
Every variable becomes a rate: dV/dt, dx/dt, dθ/dt, etc.
3. Substitute known values
only after differentiating
Plugging in numbers before differentiating is the single most common error — it treats a changing quantity as a constant.
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AP Calculus — Related Rates Practice
12 Problems
1. A spherical balloon is inflated so its radius increases at 2 cm/s. Find the rate the volume is increasing when r = 5 cm. (V = ⁴⁄₃πr³)
2. A 10 ft ladder leans against a wall. The bottom slides away at 1 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?
3. A circle's area is increasing at 4π cm²/s. Find dr/dt when r = 3 cm.
4. A point-down cone tank has radius 4 m and height 10 m. Water is pumped in at 2 m³/min. Find how fast the water level rises when the depth is 5 m.
5. Two cars leave an intersection, one north at 60 mph, one east at 80 mph. Find the rate the distance between them is increasing after 1 hour.
6. A rocket rises vertically, tracked by a camera 500 m from the pad. When the rocket is 1200 m up, its velocity is 160 m/s. Find the rate of change of the camera’s angle of elevation at that moment.
7. Sand pours onto a conical pile at 10 ft³/min, always maintaining height equal to radius. Find how fast the height increases when the pile is 5 ft high.
8. A cube's volume increases at 12 cm³/s. Find the rate its side length increases when the side is 3 cm.
9. Given xy = 100 and dx/dt = 4 when x = 10, find dy/dt at that instant.
10. A spotlight on the ground shines on a wall 12 m away. A 2 m tall man walks from the spotlight toward the wall at 1.6 m/s. Find how fast his shadow on the wall is shrinking when he is 4 m from the spotlight.
11. Water drains from a cylindrical tank of radius 3 m at 5 m³/min. Find the rate the water level is dropping.
12. If z² = x² + y², with dx/dt = 2 and dy/dt = 3, find dz/dt when x = 3 and y = 4.
The single most common error isn’t the calculus — it’s plugging in the specific numbers before differentiating. Once you substitute r = 5 into V = ⁴⁄₃πr³, you’ve frozen r as a constant, and differentiating what’s left gives dV/dt = 0. Differentiate the general equation first, then substitute the instant’s values into the resulting rate equation.
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