Model Evaporation of Raindrop with Differential Equation

[STEMucator]

Homework Statement

A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time.

Homework Equations

Volume of a sphere = V = (4/3)πr³
Surface area of a sphere = S = 4πr²

The Attempt at a Solution

So we want to write a differential equation to model the volume of the rain drop as the rain drop evaporates proportionally to its surface area over time.

So we have to consider the volume with respect to time. The volume will decrease over time, so we need a negative sign. The water evaporates at some constant rate $c$ depending on $S$.

Hence $\frac{dV}{dt} = -cS$ for some c>0 ( c must be positive otherwise the water is not evaporating ).

Unfortunately, I can't just plug $S$ in because it won't do anything useful. If I solve for the radius $r$ using $V$ and then plug my solution into $S$ I get :

$S = \sqrt[3]{9} V^{\frac{2}{3}}$

Now subbing this back I get :

$\frac{dV}{dt} = -kV^{\frac{2}{3}}$ for some k>0.

This should model the volume with respect to time as the drop evaporates.

Is my reasoning okay here or is there some things I should improve on?

 

[Ackbach]

Looks good to me!

 

[SteamKing]

S and V are all fine for an unknown shape, but you know how the volume and the surface area of the rain drop depend on the radius. IMO, your differential equation also should be written as a function of the radius with respect to time.

 

[haruspex]

S and V are all fine for an unknown shape, but you know how the volume and the surface area of the rain drop depend on the radius. IMO, your differential equation also should be written as a function of the radius with respect to time.

The requirement was to write a DE involving V and t. It would be wrong to have r in there.