rates of change: A spherical snowball is melting at a rate proportional to its surfac

[markosheehan]

A spherical snowball is melting at a rate proportional to its surface area. That is, the rate atwhich 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.

i dont know how to prove this can someone show me how to?

 

[Harry_the_cat]

A spherical snowball is melting at a rate proportional to its surface area. That is, the rate atwhich 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.

i dont know how to prove this can someone show me how to?

For a sphere:

Volume (V) = \(\displaystyle \frac{4}{3}\pi *r^3\)

Surface area (S) = \(\displaystyle 4 \pi *r^2\)

"The rate at which its volume is decreasing at any instant is proportional to its surface area at that instant" means that:
\(\displaystyle \frac{dV}{dt}=k*4\pi *r^2\) where \(\displaystyle k<0\)

Now use the chain rule: \(\displaystyle \frac{dV}{dt}= \frac{dV}{dr}*\frac{dr}{dt}\) to complete the proof.