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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?