The circumference of a circle c, is increasing at a rate of 2 cm/s, find how fast the area is increasing when r=6m?

Question

I'm having trouble solving this related rates problem. How am I supposed to find the rate at which the area changes if I am only given the rate the circumference is changing? Thanks!

Answer 1

As $c=2\pi r$, it's given that $\frac{dc}{dt}= 2\pi \frac{dr}{dt} = 2$. (given).

$A=\pi r^2$ so $\frac{dA}{dt}= 2\pi r \frac{dr}{dt}$ by the chain rule.

Now try to combine..

Answer 2

So, writing circumference as a function of the radius, $c(r) = 2\pi r$. Taking derivatives we write $\frac{dc}{dt} = 2\pi \frac{dr}{dt}$ where we are given $\frac{dc}{dt} = 2 cm/s$. Area as a function of the radius is $a(r) = \pi r^2$, and, again, taking derivatives we write $\frac{da}{dt} = \pi (2r\frac{dr}{dt})$.

You can finish this!