Chapter 5.6, Problem 2E

To determine

To find: an equation that relates $\frac{ds}{dt}$ to $\frac{dr}{dt}$ .

Answer to Problem 2E

The equation that relates $\frac{ds}{dt}$ to $\frac{dr}{dt}$ is $\frac{ds}{dt}=8\pi r\frac{dr}{dt}$ .

Explanation of Solution

Given information:

Assume all variables are differentiable function of $t$ .

  $\text{S}=4\pi r_{}^2$ ; S=Surface area and r = radius

Since all variables are differentiable function $t$ ,

Therefore,

  $\text{S}=4\pi r_{}^2$

Differentiate with respect to $t$ .

  $\frac{ds}{dt}=8\pi r\frac{dr}{dt}$

Hence, the equation that relates $\frac{ds}{dt}$ to $\frac{dr}{dt}$ is $\frac{ds}{dt}=8\pi r\frac{dr}{dt}$ .