4. A stone dropped into a lake causes circular ripples to emanate from its point of entry. The radius of the circular waves are increasing at a constant rate of 0.5 m/sec. At what rate is the circumference of a wave changing when its radius is 3 m?
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How do I solve this related rates problem? What is the solution to 3 decimal places?
![### Problem 4
A stone dropped into a lake causes circular ripples to emanate from its point of entry. The radius of the circular waves is increasing at a constant rate of 0.5 m/sec. At what rate is the circumference of a wave changing when its radius is 3 m?
#### Explanation
Given:
- The rate of change of the radius (\(dr/dt\)) is 0.5 m/sec.
- We need to find the rate of change of the circumference (\(dC/dt\)) when the radius is 3 m.
The formula for the circumference of a circle is:
\[ C = 2\pi r \]
Differentiating both sides with respect to time (\(t\)), we get:
\[ \frac{dC}{dt} = 2\pi \frac{dr}{dt} \]
Plug in the value of \(\frac{dr}{dt} = 0.5 \text{ m/sec}\) to find \(\frac{dC}{dt}\):
\[ \frac{dC}{dt} = 2\pi \times 0.5 = \pi \]
So, the rate at which the circumference is changing when the radius is 3 m is \(\pi\) meters per second.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F668826ef-a7b0-4206-bae0-a08cac22bcd4%2F214fbe6d-59c3-4cdb-8c40-722128f49b53%2F4870jse_processed.jpeg&w=3840&q=75)
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