Differential Equations - Related Rate

[kofmelk]

Homework Statement

Assume that a typical raindrop is spherical. Starting at some time, which we designate as t = 0, the raindrop of radius r sub o falls from rest from a cloud and begins to evaporate.

a) If it is assumed that a raindrop evaporates in such a manner that its shape remains spherical, then it also makes sense to assume that the rate at which the raindrop evaporates - that is, the rate at which it loses mass - is proportional to its surface area. Show that this latter assumption implies that the rate at which the radius r of the raindrop decreases is a constant. Find r(t).

There is a b part but I think I understand how that is done.

Homework Equations

Surface Area of a Sphere is 4*pi*r²

The Attempt at a Solution

-d(m)/dt is proportional to 4*pi*r²

Therefore

dm/dt = -k*4*pi*r²

Where k is a constant to remove the proportionality.

if dr/dt = constant represented by c

then after we integrate that equation we get

r(t) = c₁*t + c₂

After that I did some really weird math that I don't think is possible. Any ideas?

 

[Dick]

Mass is density (rho) times volume. So you have (rho*(4/3)*pi*r(t)^3)'=k*4*pi*r(t)^2. Can you show that implies r'(t) is constant?