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: Picture the situation.
A circular wave grows outward. Its radius is increasing, and we want how fast its area grows at the moment the radius is $10$ cm.
Step 2: Write the area formula.
The area of a circle is $A=\pi r^2$.
Step 3: Differentiate with respect to time.
Using the chain rule, $\frac{dA}{dt}=2\pi r\,\frac{dr}{dt}$.
Step 4: Note the given rates.
Here $\frac{dr}{dt}=4$ cm/sec and at the moment in question $r=10$ cm.
Step 5: Substitute the numbers.
$\frac{dA}{dt}=2\pi(10)(4)$.
Step 6: Compute.
This gives $80\pi\ \text{cm}^2/\text{sec}$, which is option (A).
\[ \boxed{\,80\pi\ \text{cm}^2/\text{sec}\,} \]