Question

A stone is thrown into a quiet lake and the waves formed move in circles. If the radius of a circular wave increases at the rate of $4\ \text{cm/sec}$, then the rate of increase in its area, at the instant when its radius is $10\ \text{cm}$, is _________ $\text{cm}^2/\text{sec}$.

• A. $80\pi$
• B. $10\pi$
• C. $8\pi$
• D. $40\pi$

Hint:

This is a standard related-rates problem. Always differentiate the geometric formula with respect to time $t$ and use the chain rule.

Answer 1

The Correct Option is A

Step 1: Write the area rule.
The wave is a circle, so its area is $A = \pi r^2$. We want how fast $A$ grows in time.

Step 2: Differentiate with respect to time.
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt}. \] This links the area rate to the radius rate.

Step 3: Put in the numbers.
Given $\tfrac{dr}{dt} = 4$ and $r = 10$: \[ \frac{dA}{dt} = 2\pi (10)(4) = 80\pi. \]

Step 4: State the rate.
\[ \boxed{80\pi\ \text{cm}^2/\text{sec}} \]