Question

salt poured from conveyor belt at 30cft/min. foms a conical pile /w a circular basewhose ht. and base diam. are equal. how fast is ht. of pile incr. when ht. is 10 ft.

Answer 1

Given: h = 2r , V = pi*r^2*h
since we want rate the height is changing, get volume in terms of height
\[V = \pi (\frac{h}{2})^{2} h = \frac{\pi h^{3}}{4}\]
Now set up a related rates equation
\[\frac{dh}{dt} = \frac{dV}{dt}*\frac{dh}{dV}\]
\[\frac{dV}{dt} = 30\]
take derivative of V wrt h
\[\frac{dV}{dh} = \frac{3 \pi h^{2}}{4} \rightarrow \frac{dh}{dV} = \frac{4}{3 \pi h^{2}}\]
thus
\[\frac{dh}{dt} = \frac{40}{\pi h^{2}}\]