# Thread: Differential equation

1. ## Differential equation

Hi, I need some help on this DEQ

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.

The back of the book says the answer: dV/dt = -kV^(2/3) for some k > 0. Though I have no idea how.

Thanks

2. Originally Posted by RB06
Hi, I need some help on this DEQ

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.

The back of the book says the answer: dV/dt = -kV^(2/3) for some k > 0. Though I have no idea how.

Thanks
Surface area of sphere:

$\displaystyle A= 4 \pi r^2$

Volume of sphere:

$\displaystyle V = \frac{4}{3} \pi r^3$

Now you are told that:

$\displaystyle \frac{dV}{dt} = c A$

for some constant $\displaystyle c$, but $\displaystyle A \propto V^{2/3}$, so:

$\displaystyle \frac{dV}{dt} = k V^{2/3}$

for some constant $\displaystyle k$

RonL