- Topic
- Related rates
- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 4
Related rates practice question 22
Gravel falls from a conveyor onto a conical pile whose height is always three quarters of the diameter of its base. Gravel is added at 10 m³/min. How fast is the height of the pile increasing at the moment the pile is 3 m high?
Give the answer in metres per minute.
The height is tied to the diameter, 2r, so the radius is tied to the height. Use that to write V in terms of h alone before differentiating.
, so and .
A rate in m/min, exact or to two decimal places.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
The shape of the pile ties its radius to its height, so the volume is a function of the height alone. Eliminate r, differentiate with respect to time, and solve for dh/dt at the given height.
Working
The diameter is 2r, and the ratio holds at every instant.
Volume in terms of the height alone.
Differentiate with respect to time.
Substitute h = 3 and dV/dt = 10 only now.
Answer
The answer is 5 over 2 pi metres per minute, about 0.796.
Check
A pile of fixed shape scales with the cube root of its volume, so 0.1 m³ more on 4π ≈ 12.57 m³ lifts it from 3 to about 3.0079 m in 0.01 min.
Where this goes wrong
Putting in r = 2, the radius when the pile is 3 m high, before differentiating. That freezes the width of the pile and gives 15/(2π). Eliminate r with r = 2h/3 first, differentiate, and substitute h = 3 last.
Leans on
Similar triangles, from Functions, graphs and inverses
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