Concentric circles form when a stone is dropped into a pool of water
a) What is the average rate of change in the area of one circle with respect to the radius as the radius grows from 0 cm to 100 cm? (Answer: $100\pi \frac{cm^2}{cm}$)
b) How fast is the area changing with respect to the radius when the radius is 120 cm?
My attempt:
$A$ is area, $r$ is radius.
$$\frac{\Delta A}{\Delta r}=\frac{\pi(120)^2-\pi(100)^2}{120-100}$$
$$=\frac{14400\pi-10000\pi}{20}$$
$$=\frac{14400\pi}{20}-\frac{10000\pi}{20}$$
$$=720\pi-500\pi$$
$$=220\pi$$
The problem is that my text says the answer is $240\pi$, so I've gone wrong somewhere. Any help?
Best Answer
What you calculated is the average rate of change of area between radii of 100 and 120 cm, not the instantaneous rate at 120 cm.
To solve this problem, note that $\frac d{dr}\pi r^2$ (derivative of the circle's area with respect to its radius) is $2\pi r$, and substituting $r=120$ yields the correct answer of $240\pi$.