# Derivatives, rates of change (cone)

1. Dec 2, 2013

### physics604

1. Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

2. Relevant equations
$$V=\frac{\pi}{3}r^2h$$
3. The attempt at a solution

Diameter = height, so $$\frac{h}{2}=r$$
$$V=\frac{\pi}{3}\frac{h^2}{4}h = \frac{\pi}{12}h^3$$
$$\frac{dV}{dt}=\frac{\pi}{12}(3×h)\frac{dh}{dt}$$
$$30=\frac{\pi}{12}(3×10)\frac{dh}{dt}$$
$$\frac{dh}{dt}=\frac{12}{\pi}$$

The textbook's answer is $$\frac{6}{5\pi}$$ What did I do wrong?

Last edited: Dec 2, 2013
2. Dec 2, 2013

### SteamKing

Staff Emeritus
You incorrectly assumed that the diameter of the pile = half of the radius.

3. Dec 2, 2013

### physics604

Sorry that was just a typo. The work should still follow r=h/2.

4. Dec 2, 2013

### SteamKing

Staff Emeritus