A stone is dropped into a quiet lake and waves move in circles. The radius of a circular wave increases at the rate of 3 cm/s. The rate at which the area enclosed by the wave is increasing when the radius is 10 cm is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(60\pi\) cm2/s
Let r be the radius and A the area enclosed by the wave at time t. Then
\(A = \pi r^{2}\)
Differentiating with respect to t by the chain rule:
\( \displaystyle \frac{dA}{dt} = 2\pi r\frac{dr}{dt} \)
With r = 10 and \(\frac{dr}{dt} = 3\):
\( \displaystyle \frac{dA}{dt} = 2\pi \times 10 \times 3 \)
\( = 60\pi \) cm2/s
Why the other options are incorrect
Option (1) \(30\pi\) cm2/s — Incorrect. This is \(\pi r \times 3\): the factor 2 from differentiating \(r^{2}\) is missing.
Option (2) \(6\pi\) cm2/s — Incorrect. This is \(2\pi \times 3\), the rate at which the circumference \(2\pi r\) increases, not the area.
Option (4) \(100\pi\) cm2/s — Incorrect. This is the area \(\pi \times 10^{2}\) itself at that instant, not its rate of change.