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To solve this problem, we need to determine the rate at which the area of the ripple (circular wave) is changing when the radius is 12 feet. Here’s how to solve it:

### Given Data:
- The radius \( r \) of the ripple is increasing at a rate of \( \frac{dr}{dt} = 5 \) feet per second.
- We are asked to find the rate of change of the area \( A \) at the moment when \( r = 12 \) feet.

### Solution:
The area \( A \) of a circle with radius \( r \) is given by:
\[
A = \pi r^2
\]

To find how fast the area is changing with respect to time, we need to differentiate \( A \) with respect to \( t \):
\[
\frac{dA}{dt} = \frac{d}{dt} (\pi r^2)
\]

Using the chain rule, this becomes:
\[
\frac{dA}{dt} = 2\pi r \frac{dr}{dt}
\]

### Substitute the Known Values:
- When \( r = 12 \) feet and \( \frac{dr}{dt} = 5 \) feet per second:

\[
\frac{dA}{dt} = 2\pi (12)(5)
\]

\[
\frac{dA}{dt} = 120\pi \quad \text{square feet per second}
\]

### Approximate the Result:
To find the approximate value, we can use \( \pi \approx 3.14159 \):

\[
\frac{dA}{dt} \approx 120 \times 3.14159 \approx 376.991 \quad \text{square feet per second}
\]

### Final Answer:
The rate at which the area is changing when the radius is 12 feet is approximately **376.991 square feet per second**.

Would you like more details on any part of this solution or have any questions?

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Here are some related questions for further exploration:
1. What happens to the rate of change of the area as the radius increases?
2. How would the solution change if the rate of radius change varied over time?
3. What is the relationship between the radius and the circumference rate of change in this context?
4. How would you calculate the rate of change of the circumference instead of the area?
5. Can you find a general formula for \( \frac{dA}{dt} \) in terms of \( r \) and \( \frac{dr}{dt} \)?

**Tip:** Understanding how the chain rule applies in related rates problems is crucial, as it helps link different quantities that change over time.

A rock is thrown into a still pond. The circular ripples move outward from the point of impact of the rock so that the radius of the circle formed by a ripple increases at the rate of 5 feet per second. Find the rate at which the area is changing at the instant the radius is 12 feet.

Learn how to solve a related rates problem involving circular ripples in a pond. This problem calculates the rate of change of the area when the radius of the circle is increasing at 5 feet per second. Suitable for high school calculus students.

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