# What is the rate of change of the area of

What is the rate of change of the area of a circle with respect to the radius when the radius is r = 3 in.?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Virginia Palmer
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:
$\underset{h\to 0}{lim}\frac{\pi {\left(r+h\right)}^{2}-\pi {r}^{2}}{h}=\underset{h\to 0}{lim}\frac{\pi \left({r}^{2}+2hr+{h}^{2}\right)-\pi {r}^{2}}{h}$
$=\underset{h\to 0}{lim}\frac{\pi \left(2hr+{h}^{2}\right)}{h}$
$=\underset{h\to 0}{lim}\frac{\pi h\left(2r+h\right)}{h}$
$=\underset{h\to 0}{lim}\left[\pi \left(2r+h\right)\right]$
Substitute $r=3$:
$=\underset{h\to 0}{lim}\left[\pi \left(6+h\right)\right]$
$=6\pi$
When the radius is 3 in., area is changing at a rate of $6\pi$