Question

Help needed in related rates of change in the following problem:

The area of a circle is decreasing at the rate of 2cm^2s^-1. How fast is the radius decreasing when the area is 9*pi

I've tried finding the derivative of A with respect to r (dA/dr) and inversing it to get dr/dA didn't work either . Taking dr/dA directly didn't work either.

Answer 1

\[ A= \pi r^2 \\ \frac{dA}{dr} = 2\pi r\\ dA= 2 \pi r \ dr \]
from the given info we know
\[ A= \pi r^2 \\ 9 \pi = \pi r^2 \]
so we can find r
we also know dr = 2 cm
so we can find dA

Answer 2

strictly speaking the ratio of infinitesimals such as \( \frac{dA}{dr} \) cannot be split like an ordinary fraction. what we do is say, the approximate differential is
\[ \frac{\Delta A}{\Delta r}\]
where we have a ratio of "tiny" but measurable quantities. Now it is ok to multiply both sides by \( \Delta r \)
and we have the approximation
\[ \Delta A \approx 2 \pi r \Delta r \]