To solve this problem, let's break it down step by step:

Given:

• The radius $r$ of the circular ripples increases at a rate of $0.2$ meters per second.
• We need to find the value of $A \pi$ after 5 seconds, where $A$ is the area of the circle.

Step 1: Find the radius after 5 seconds.

The rate of change of the radius is given by:

$\frac{dr}{dt} = 0.2 \, \text{m/s}$

This means that the radius increases by $0.2$ meters every second. Since 5 seconds have passed, the radius after 5 seconds is:

$r = 0.2 \, \text{m/s} \times 5 \, \text{s} = 1 \, \text{meter}$

Step 2: Find the area of the circle after 5 seconds.

The formula for the area $A$ of a circle is:

$A = \pi r^2$

Substituting $r = 1$ meter:

$A = \pi \times (1)^2 = \pi \, \text{square meters}$

Step 3: Find $A \pi$.

You want the value of $A \pi$, which is:

$A \pi = \pi \times \pi = \pi^2$

So, the value of $A \pi$ is $\pi^2$, which is approximately:

$A \pi \approx 9.87 \, \text{square meters}$

Final Answer:

The value of $A \pi$ after 5 seconds is $\pi^2$, or approximately $9.87 \, \text{m}^2$.

Would you like more details on the solution, or have any questions about this?

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Here are 5 follow-up questions that build on the topic:

1. How does the rate of change of the radius affect the area of the circle?
2. What would happen to the value of $A \pi$ if the radius increased at a faster rate, say 0.3 meters per second?
3. How would the area change if the radius increased at a rate of $0.1$ meter per second for 10 seconds?
4. How does the concept of related rates apply in this problem?
5. How would you generalize this formula to find $A \pi$ for any given time?

Tip: When solving related rates problems, always start by identifying the rate of change and writing down the formulas you need to use.