The area of the oil spill is given by the formula $A = \pi t^2$, where $A$ is the area in square meters and $t$ is the time in minutes.

The average rate of change of the area over a time interval is found using the formula:

$\text{Average rate of change} = \frac{A(t_2) - A(t_1)}{t_2 - t_1}$

Here, $t_1 = 0$ (the initial time) and $t_2 = 5$ (the final time after 5 minutes).

Step 1: Calculate $A(0)$

At $t = 0$: $A(0) = \pi (0)^2 = 0$

Step 2: Calculate $A(5)$

At $t = 5$: $A(5) = \pi (5)^2 = 25\pi$

Step 3: Find the average rate of change

Now, using the formula: $\text{Average rate of change} = \frac{A(5) - A(0)}{5 - 0} = \frac{25\pi - 0}{5} = 5\pi \text{ square meters per minute}$

Thus, the average rate of change of the area of the oil spill over the first 5 minutes is approximately $5\pi$ square meters per minute, or about $15.71$ square meters per minute (using $\pi \approx 3.1416$).

Would you like further clarification on this or any additional details?

Here are 5 related questions:

1. What is the instantaneous rate of change of the area at $t = 5$ minutes?
2. How would the average rate of change differ if the time interval was from 2 to 6 minutes?
3. What is the physical meaning of the rate of change in this context?
4. How would the formula for area change if the oil spill expanded at a rate proportional to $t^3$ instead of $t^2$?
5. How would you calculate the area of the oil spill after 10 minutes?

Tip: In calculus, the average rate of change over an interval is analogous to the slope of the secant line between two points on a curve.