physics604
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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?
$$V=\frac{\pi}{3}r^2h$$
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?
Homework Equations
$$V=\frac{\pi}{3}r^2h$$
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?
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