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: Understanding the Question:
We need to find the rate of change of the area of a circle with respect to time, given the rate of change of its radius.

Step 2: Key Formula or Approach:
The area of a circle is $A = \pi r^2$. Differentiate with respect to $t$: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$.

Step 3: Detailed Explanation:
Given: $\frac{dr}{dt} = 4\ \text{cm/sec}$ and $r = 10\ \text{cm}$.
Using the derivative formula:
$\frac{dA}{dt} = 2 \cdot \pi \cdot (10\ \text{cm}) \cdot (4\ \text{cm/sec})$
$\frac{dA}{dt} = 80\pi\ \text{cm}^2/\text{sec}$.

Step 4: Final Answer:
The rate of increase in area is $80\pi\ \text{cm}^2/\text{sec}$, which corresponds to option (A).