Equation for a Sphere with Increasing Volume

Question

The Problem:

A sphere with radius $r$ increases in volume at a rate proportional to the surface area of the sphere at that time, with proportionality constant $k$. Write a function for the sphere’s volume at any time $t$.

My Attempt:

We can write an ODE to solve for $r(t)$, and then plug that into $V=\frac{4}{3}\pi r^3$. Since the rate of change of the radius is equal to the surface area $4\pi r^2$ times $k$, we get:$$\frac{dr}{dt}=4\pi kr^2$$

I separated the variables and got:$$r(t)=-\frac{1}{4\pi kt+C}$$Let alone the volume function, this radius function makes no sense. Help please!

Answer 1

You've gotten the main equation wrong: it should be $\dfrac{d\color{red}V}{dt}=4\pi kr^2$, not $\dfrac{d\color{red}r}{dt}=4\pi kr^2$. By the chain rule, we get $\dfrac{dV}{dt}=\dfrac{dV}{dr}\dfrac{dr}{dt}$. But from the known relation $V=\dfrac43\pi r^3$, we have $\dfrac{dV}{dr}=4\pi r^2$. Substituting this we get $\dfrac{dr}{dt}=k$, from where $r=kt+C$, and $C=r(0)$.

Answer 2

$$\frac d{dt}\left(\frac43\pi r^3\right)=k\,4\pi r^2\iff\frac{dr}{dt}=k\iff r(t)=kt+r(0).$$