Melting Snowball: A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface Area =4(pi)r^2)

Answer 1

dV/dt = kA.

V = (4/3) π r^3 and A = 4 π r^2

dV/dt = d/dt((4/3) π r^3) = (4/3) π 3 r^2 (dr/dt)

Now let's plug that into the first equation:

(4/3) π 3 r^2 (dr/dt) = k A = k(4 π r^2) = 4 π k r^2

So when we simplify by dividing left and right sides by 4 π r^2, we get:

dr/dt = k

which is saying exactly what we wanted to prove: that dr/dt is constant.

Answer 2

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