Question

The radius of a circle is increasing at a rate of 6 centimeters per minute. Find the rate of change of the area when the radius is 4 centimeters.

Answer 1

Answer:

The rate of change is of of the area when the radius is 4 centimeters $48\pi$ square centimeters.

Step-by-step explanation:

Area of a circle:

The area of a circle of radius r is given by:

$A = \pi r^2$

Implicit derivative:

To solve this question, we have to derivate implictly the equation as a function of t. Thus:

$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$

The radius of a circle is increasing at a rate of 6 centimeters per minute.

This means that $\frac{dr}{dt} = 6$

Find the rate of change of the area when the radius is 4 centimeters.

This is $\frac{dA}{dt}$ when $r = 4$. Thus

$\frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 2\pi(4)(6) = 48\pi$

The rate of change is of of the area when the radius is 4 centimeters $48\pi$ square centimeters.