The area, A, in square metres, of a circular oil spill after t min is given by A = pt2.
The average rate of change of the area of the oil spill over the first 5 min is approximately

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.

The area, A, in square metres, of a circular oil spill after t min is given by A = πt^2. The average rate of change of the area of the oil spill over the first 5 min is approximately.

Calculate the average rate of change of a circular oil spill expanding with the formula A = πt^2 over the first 5 minutes. This problem explores key algebraic concepts such as quadratic growth and average rate of change, ideal for students in Grades 10-12.

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