MHB Proof That Radius of Melting Snowball Decreases Constantly

[markosheehan]

A spherical snowball is melting at a rate proportional to its surface area. That is, the rate at
which its volume is decreasing at any instant is proportional to its surface area at that instant.
(i) Prove that the radius of the snowball is decreasing at a constant rate.

can someone help me?

 

[HallsofIvy]

A spherical snowball is melting at a rate proportional to its surface area. That is, the rate at
which its volume is decreasing at any instant is proportional to its surface area at that instant.
(i) Prove that the radius of the snowball is decreasing at a constant rate.

can someone help me?

Let V be the volume and S the surface area: dV/dt= kS for some constant k.
Now, You need to know that $V= \frac{4}{3}\pi r^3$ and $S= 4\pi r^2$.

So dV/dt= kS becomes $\frac{d(\frac{4}{3}\pi r^3)}{dr}= k(4\pi r^2)$.
Simplify the left side.