Question

Find the rate of change of the area of a circle with respect to its radius r when

r = 4 cm

Answer 1

Let A denotes the area of the circle when its radius is r, then $A=\pi r^2$
Now, the rate of change of the area with respect to its radius is given by,
$\frac{dA}{dr}=\frac{d}{dr}\left(\pi r^2\right)=2\pi r$
When $r=4cm,{\left(\frac{dA}{dr}\right)}_{r=4}=2\pi \left(4\right)=8\pi c{m}^2$ per cm
Hence, the area of the circle is changing at the rate of $8\pi c{m}^2/cm$
when its radius is 4 cm.