Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop's radius increases at a constant rate. A sphere with radius r has volume V = and surface area S =]. Find - dv dt dV From the problem statement, dV it is proportional to the surface area of the sphere. This means that if k is a constant, then dt -= dV Substitute for and S in the equation from the previous step and interpret the results. The radius increases at a constant rate because the equation simplifies to + = , a constant