Question:
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
(A) $10 \pi$ (B) $12 \pi$ (C) $8 \pi$ (D) $11 \pi$
Solution:
The area of a circle (A) with radius (r) is given by,
$A=\pi r^{2}$
Therefore, the rate of change of the area with respect to its radius r is
$\frac{d A}{d r}=\frac{d}{d r}\left(\pi r^{2}\right)=2 \pi r$
$\therefore$ When $r=6 \mathrm{~cm}$,
Hence, the required rate of change of the area of a circle is $12 \pi \mathrm{cm}^{2} / \mathrm{s}$.
The correct answer is B.