JAMB Mathematics 2023 Paper

Concept:Use differentiation and the chain rule to connect the rate of change of area to the rate of change of radius.Explanation:The area of a circle is $A = \pi r^2$.Differentiate $A$ with respect to $r$:$\frac{dA}{dr} = 2\pi r$.The area increases at $\frac{dA}{dt} = 1.5\,\text{cm}^{2}\text{s}^{-1}$.We need $\frac{dr}{dt}$ when $A = 2\,\text{cm}^{2}$.First, find the radius at that instant:$2 = \pi r^2$, so $r = \sqrt{\frac{2}{\pi}} \approx 0.798\,\text{cm}$.By the chain rule, $\frac{dr}{dt} = \frac{dA}{dt} \times \frac{dr}{dA}$.From the derivative above, $\frac{dr}{dA} = \frac{1}{2\pi r}$.Therefore, $\frac{dr}{dt} = 1.5 \times \frac{1}{2\pi(0.798)}$.Since $2\pi(0.798) \approx 5.014$, we get $\frac{dr}{dt} \approx 1.5 \times 0.1994$.$\frac{dr}{dt} \approx 0.299\,\text{cm}\,\text{s}^{-1}$ to 3 significant figures.Answer:The rate at which the radius is increasing is $0.299\,\text{cm}\,\text{s}^{-1}$, so the correct option is D.