A stone drops into a pond, creating a circular ripple - HSC - SSCE Mathematics Extension 1 - Question 8 - 2017 - Paper 1

To find the rate of change of the area with respect to time, we apply the chain rule of differentiation:

$\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}$

Calculating the derivative of the area with respect to the radius:

$\frac{dA}{dr} = 2\pi r$

Given that the radius r is increasing at a rate of 5 cm s⁻¹, we substitute:

• $r = 15 \text{ cm}$
• $\frac{dr}{dt} = 5 \text{ cm s}^{-1}$

So, $\frac{dA}{dt} = 2\pi (15) (5)$

Now, calculating the above expression:

$\frac{dA}{dt} = 2\pi (15) (5) = 150\pi \text{ cm}^2 \, \text{s}^{-1}$

Thus, the area is increasing at a rate of $150\pi \text{ cm}^2 \, \text{s}^{-1}$.

The correct answer is C. $150\pi \text{ cm}^2 \, \text{s}^{-1}$.