"Helium is pumped into a spherical balloon at a rate of $4$ cubic feet per second. How fast is the radius increasing after $3$ minutes?"
So this is what I did:
$$V = \frac{4}3\pi r^3$$
$$\frac{dv}{dt} = 4\pi r^2\frac{dr}{dt}$$
$$4 = 4\pi r^2\frac{dr}{dt}$$
$$\frac{dr}{dt} = \frac{4}{4\pi r^2}$$
I did not know how to proceed after this. How am I supposed to find the value of $r$? Do I have to somehow relate $3$ minutes ($180$ seconds) in somehow?
Any help?
The volume is increasing at a constant rate.
$$\frac{dV}{dt}=4$$
and also $V(0)=0$.
Hence, at time $t=3$, you can compute the volume at that time and you can use the formula
$$V=\frac{4}{3}\pi r^3$$
to solve for $r$ at that time.