Chapter 5.6, Problem 1E

To determine

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

Answer to Problem 1E

The equation that relates $\frac{d\text{A}}{dt}$ to $\frac{dr}{dt}$ is $\frac{d\text{A}}{dt}=2\pi r\frac{dr}{dt}$ .

Explanation of Solution

Given information:

All variables are differentiable function of $t$ .

  $\text{A}=\pi r_{}^2$ ; A = area, r = radius

Calculation :

We have to relate $\frac{d\text{A}}{dt}$ to $\frac{dr}{dt}$

Since all variables are differentiable function of $t$

Therefore,

  $\begin{array}{l}\text{A}=\pi r_{}^2\\ \frac{d\text{A}}{dt}=2\pi r\frac{dr}{dt}\end{array}$

Hence, the equation that relates $\frac{d\text{A}}{dt}$ to $\frac{dr}{dt}$ is $\frac{d\text{A}}{dt}=2\pi r\frac{dr}{dt}$ .