Josh Sizemore
2021-12-24
Answered

What is the rate of change of the area of a circle with respect to the radius when the radius is r = 3 in.?

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Virginia Palmer

Answered 2021-12-25
Author has **27** answers

The area of a circle is given as

$A\left(r\right)=\pi {r}^{2}$

To find the instantneous rate of change with respect to r, we apply the slope formula:

$\underset{h\to 0}{lim}\frac{\pi {(r+h)}^{2}-\pi {r}^{2}}{h}=\underset{h\to 0}{lim}\frac{\pi ({r}^{2}+2hr+{h}^{2})-\pi {r}^{2}}{h}$

$=\underset{h\to 0}{lim}\frac{\pi (2hr+{h}^{2})}{h}$

$=\underset{h\to 0}{lim}\frac{\pi h(2r+h)}{h}$

$=\underset{h\to 0}{lim}\left[\pi (2r+h)\right]$

Substitute$r=3$ :

$=\underset{h\to 0}{lim}\left[\pi (6+h)\right]$

$=6\pi$

When the radius is 3 in., area is changing at a rate of$6\pi$

To find the instantneous rate of change with respect to r, we apply the slope formula:

Substitute

When the radius is 3 in., area is changing at a rate of

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