# Math Help - Rate of increase on conical pile

1. ## Rate of increase on conical pile

Sand is being dumped at a constant rate onto a conical pile, which is 5 feet high. When you see it again, 1 hour and 10 minutes later, the pile is 10 feet high. At what rate is the height increasing?

If you can help me figure out this problem I'd appreciate it

2. We may first write all of the information given in mathematical notation:

\begin{aligned}
V&=\frac{1}{3}\pi r^2h\\
h(0)&=5\\
h(70)&=10\\
\frac{dV}{dt}&=C.
\end{aligned}

In order that the problem have an answer, the ratio of the height to the radius, and thus the angle of the tip of the cone, must remain the same. (Otherwise, we could have the cone start out narrow and then get wider as $h$ approaches $10$, lowering our answer for $\frac{dh}{dt}$ dramatically.) Therefore, we may say that

$r=hk$

for some $k$, and that

$V=\frac{1}{3}\pi(hk)^2h=\frac{k^2\pi}{3}h^3.$

In order to find the value of $\frac{dh}{dt}$ at $t=70$, in minutes, we may use the equations above together with the fact that

$\frac{V(70)-V(0)}{70}=C.$