1. ## Composite function problem

Here is the problem I'm trying to solve:

The volume of a conical pile of sand is increasing at a rate of 243(pi)ft^3/min, and the height of the pile always equals the radius r of the base. Express r as a function of time t (in minutes), assuming that r = 0 when t = 0.

2. Hello, Tubaman!

$\text{The volume of a conical pile of sand is increasing at }243\pi\text{ft}^3\!/\text{min}$
$\text{The height }h\text{ of the pile always equals the radius }r\text{ of the base.}$
$\text}Express }r\text{ as a function of time }t\text{, assuming that }r\,\!=\,\!0 \text{ when }t\,\!=\,\!0.$

The volume of a cone is: . $V \;=\;\frac{\pi}{3}r^2h$

Since $r = h$, we have: . $V \;=\;\frac{\pi}{3}r^3$

Differentiate with respect to time: . $\dfrac{dV}{dt} \;=\;\pi r^2\dfrac{dr}{dt}$

We are told that: $\dfrac{dV}{dt} \,=\,243\pi\text{ ft}^3\!/\text{min}$

Hence: . $\pi r^2\dfrac{dr}{dt} \;=\;243\pi \quad\Rightarrow\quad r^2\,dr \;=\;243\,dt$

Integrate: . $\frac{1}{3}r^3 \;=\;243t + c \quad\Rightarrow\quad r^3 \;=\;729t + C$

When $r=0,\:t=0\!:\;\;0^3 \:=\:729(0) + C \quad\Rightarrow\quad C \,=\,0$

So we have: . $r^3 \:=\:729t$

Therefore: . $r \;=\;9\sqrt[3]{t}$