Part A

Take the volume formula for a sphere and solve it for r.

Then determine a formula for surface area in terms of V(t). We then substitute this into our differential equation model which says dV/dt is directly proportional to the capsule's surface area.

This yields a differential equation in V[t].

Which we solve and whose constant of integration we determine in terms of the initial condition.

And hence we have a model of the volume of the capsule as a function of time t.