The formula for the area of the oil spill is given as:

$A = p t^2$

To find the instantaneous rate of change of the area after 2 minutes, we need to compute the derivative of $A$ with respect to $t$, and then evaluate it at $t = 2$.

1. First, take the derivative of $A$ with respect to $t$:

$\frac{dA}{dt} = \frac{d}{dt} \left( p t^2 \right) = 2p t$

2. Now, substitute $t = 2$ into the derivative to find the rate of change at that specific time:

$\frac{dA}{dt} \Bigg|_{t=2} = 2p(2) = 4p$

Thus, the instantaneous rate of change of the area of the oil spill after 2 minutes is approximately $4p$ square meters per minute.

Would you like further details or have any questions?

Here are 5 related questions:

1. How does the formula for area change if the spill spreads at a variable rate?
2. What is the interpretation of the derivative in this context?
3. How would the rate of change differ if $t = 3$ minutes?
4. Can we interpret the constant $p$ in terms of physical properties of the oil?
5. How would the analysis change if the relationship between area and time was not quadratic?

Tip: The derivative gives the rate of change at any instant, which is crucial for understanding how a system evolves over time.