Question

Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm.

Answer 1

Let A be the area of the circular disc. Then,

A = $\mathrm{\pi }{r}^2$

$\Rightarrow \frac{dA}{dr}=2\mathrm{\pi }r$
Let C be the circumference of the circular disc. Then,

C = $2\mathrm{\pi }r$

$\Rightarrow \frac{dC}{dr}=2\mathrm{\pi }$

$$\begin{gathered} \therefore \frac{dA}{dC}=\frac{{\displaystyle \raisebox{1ex}{$dA$}\!\left/ \!\raisebox{-1ex}{$dr$}\right.}}{{\displaystyle \raisebox{1ex}{$dC$}\!\left/ \!\raisebox{-1ex}{$dr$}\right.}} \\ \Rightarrow \frac{dA}{dC}=\frac{2\mathrm{\pi }r}{2\mathrm{\pi }}=r \\ \Rightarrow {\left(\frac{dA}{dC}\right)}_{r=3}=3\mathrm{cm} \end{gathered}$$