Find the rate of change of the area of a circle per second with respect to its radius $r$ when $r = 5 \text{ cm}$.

$10\pi \text{ cm}^2/\text{cm}$

The correct answer is Option (2) → $10\pi \text{ cm}^2/\text{cm}$ ##

The area $A$ of a circle with radius $r$ is given by $A = \pi r^2$. Therefore, the rate of change of the area $A$ with respect to its radius $r$ is given by

$\frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r$

When $r = 5 \text{ cm}$, $\frac{dA}{dr} = 10\pi$. Thus, the area of the circle is changing at the rate of $10\pi \text{ cm}^2/\text{s}$.