# Spherical raindrop problem

## Homework Statement

A spherical raindrop evaporates at a rate proportional to its surface area with (positive) constant of proportionality k; i.e. the rate of change of the volume exactly equals −k times the surface area. Write differential equations for each of the quantities below as a function of time. For each case the right hand side should be a function of the dependent variable and the constant k. For example, the answer to the first question should not depend on S or r.

## Homework Equations

I was able to find dV/dt, dr/dt using mathematical models but I can't figure out why my answer isn't right for dS(surface area)/dt.

## The Attempt at a Solution

My dV/dt is -k\left(36\pi \right)^{\left(\frac{1}{3}\right)}V^{\left(\frac{2}{3}\right)}
My dr/dt is -k.
Because the equation for the surface area is 4*pi*r^2, the derivative of this would be 8*pi*r dr/dt.
Thus, this can be rewritten as -8k*pi*r. However, because my right side cannot include any independent variables, I must write by r in terms of Volume, which is r = (3V/4pi)^(1/3). So I put down my answer using these but in turns out its wrong so I'm kinda lost.

Here's my second attempt. My s is s = ((36pi)^(1/3))*V^(2/3). Taking the derivative of this would be {(2((36pi)^(1/3))*V^(-1/3))/3}*(dv/dt). And I think plugging in dv/dt would give me a solution but if that's the case, I don't know why my answer would differ.

Also, if there's any way to put down formulas without using brackets and stuff, would much appreciate it if someone can tell me

Orodruin
Staff Emeritus