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:

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

First, differentiate A with respect to r:

$\frac{dA}{dr} = 2\text{π} r$

Now substitute this into the equation:

$\frac{dA}{dt} = 2\text{π} r \cdot \frac{dr}{dt}$

Given that the radius r is 15 cm and the rate of change of radius (\frac{dr}{dt}) is 5 cm s⁻¹:

$\frac{dA}{dt} = 2\text{π} (15) (5)$

Calculate the value:

$\frac{dA}{dt} = 150\text{π} \text{ cm}^2 ext{s}^{-1}$