Chapter 5.6, Problem 37E

To determine

To justify: whether the statement is true or false.

Answer to Problem 37E

The statement is false.

Explanation of Solution

Given information :

If the radius of a circle is expanding at a constant rate, then its area is increasing at a constant rate.

All variables are differentiable functions of t .

Calculation:

Let,

  $\frac{d\text{R}}{dt}=x\left[\text{R}=\text{Radius of a circle}\text{.}\right]$

Where x is a constant rate of expanding then area of a circle is

  $\text{A}=\pi {\text{R}}^2$

Differentiate with respect to t using product rule.

  $\frac{d\text{A}}{dt}=2\pi \text{R}\frac{d\text{R}}{dt}$

Putting the value of $\frac{d\text{R}}{dt}=x$ in above equation,

  $\frac{d\text{A}}{dt}=2\pi \text{R}x$

Hence, the area is increasing at an increasing rate of $2\pi \text{R}x$ .