Ratio comparing rate of volume of sphere and surface area

Question

Find and express a ratio that compares the rate at which the volume of the sphere changes with respect to the rate at which the surface area of the sphere changes (the ratio will be a function of $r$). I have found what the change in the volume with respect to time ($4\pi(r)^2*dr/dt$) and the change in the surface area with respect to time ($8\pi(r)*dr/dt$). I am confused on how I should write this ratio and if I should write it with respect to $r$ or $t$. Any help is appreciated!

Answer 1

You already have your rates. The ratio of two things is just one thing divided by the other thing. So take your rate for the change in volume and divide it by the change in area.

If you write the division in the format shown below, $$ \frac{\text{rate of change of volume}}{\text{rate of change of area}}, $$ then you may notice some opportunities to cancel factors in the numerator and denominator. Note that $dr/dt$ is just another number when you’re comparing the two rates at the same time. (If you were looking at $dr/dt$ at two different times, you should not assume it stayed the same, but we can be reasonably sure we were supposed to compare two things at the same time; the question makes no sense otherwise.)