Related Rates Question

Question

The question reads:

The radius of a circle is decreasing at a constant rate of 0.1 centimeters per second. In terms of circumference, C, what is the rate of change of the area, A, of the circle, in square centimeters per second?

The right answer goes this way:

$ A = \pi r^2 $ --> $ \frac{da}{dt}=2\pi r \frac{dr}{dt} = 2\pi \frac {c}{2\pi} \frac{dr}{dt} = C*-.1 $

Why do I get the wrong answer this way: $ \frac{da}{dt}=2\pi (\frac{C}{2\pi}) \frac{dC}{dt} = C * -.2\pi $ Does not $\frac{dC}{dt}= -.2\pi$ if$ \frac{dr}{dt} = -.1$ in the example?

Answer 1

It is simply the case that

$$\frac{dr}{dt}\ne\frac{dC}{dt}$$

Clearly, they differ by a coefficient of $2\pi$. It is true that $\frac{dC}{dt}=-0.2\pi$, but that is not related to the problem. Also, I would do the problem as follows:

$$\frac{dA}{dt}=\frac d{dt}\pi r^2=\left(\frac{dr}{dt}\right)\cdot(2\pi r)=-0.1C$$

which follows from $\frac d{dt}r^2=2r\frac{dr}{dt}$ by chain rule, $\frac{dr}{dt}$ is given, and $C=2\pi r$.