To solve this problem, let's denote the variables given and find the rate at which the area inside the circular wave is changing:

1. Let $r$ be the radius of the circle, which is increasing at a rate of $\frac{dr}{dt} = 3$ feet per second.
2. The formula for the area $A$ of a circle is $A = \pi r^2$.

We need to find the rate of change of the area with respect to time, $\frac{dA}{dt}$, when $r = 5$ feet.

Steps

1. Differentiate the area formula with respect to time $t$: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$

2. Substitute $r = 5$ feet and $\frac{dr}{dt} = 3$ feet per second into the equation: $\frac{dA}{dt} = 2\pi (5)(3) = 30\pi \text{ square feet per second}$

So, the rate at which the area inside the circular wave is changing when the radius is 5 feet is $30\pi$ square feet per second.

Would you like further details on any part of this solution?

Here are 5 related questions to delve deeper:

1. What would the rate of area change be if the radius were 10 feet instead of 5?
2. How would the solution change if the rate of radius increase were different, say 4 feet per second?
3. How does the relationship between area and radius apply in real-world circular wave phenomena?
4. What other methods could be used to solve this problem using calculus concepts?
5. Can this concept be applied to non-circular shapes, and how would that affect the solution?

Tip: When dealing with related rates, always write down what rates you know and what you need to find, then differentiate with respect to time.