To find an exponential model for the area $A$ of the oil spill as a function of time $t$, let's work through the information provided:

1. Initial Area: The initial area of the oil spill, when $t = 0$, is 2 square miles.
2. Growth Rate: The area triples every 6 hours.

Step-by-Step Solution

1. Model Form: The general form for exponential growth is: $A(t) = A_0 \cdot e^{kt}$ where:

   • $A(t)$ is the area of the spill at time $t$,
   • $A_0$ is the initial area,
   • $k$ is the growth constant,
   • $t$ is the time in hours.

2. Initial Condition: Since $A(0) = 2$, we know: $A(0) = 2 \Rightarrow A(t) = 2 \cdot e^{kt}$

3. Determine $k$ Using the Growth Rate: We know the area triples every 6 hours. This means: $A(6) = 3 \times A(0) = 3 \times 2 = 6$

   Substitute $t = 6$ and $A(6) = 6$ into the model: $6 = 2 \cdot e^{6k}$

4. Solve for $k$: Divide both sides by 2: $3 = e^{6k}$ Take the natural logarithm of both sides: $\ln(3) = 6k$ Solve for $k$: $k = \frac{\ln(3)}{6}$

5. Final Model: Substitute $k$ back into the model: $A(t) = 2 \cdot e^{\left(\frac{\ln(3)}{6}\right) t}$

Simplified Model

Thus, the exponential model for the area of the oil spill as a function of time $t$ in hours is: $A(t) = 2 \cdot e^{\left(\frac{\ln(3)}{6}\right) t}$

This model describes the growth of the oil spill area over time based on the given conditions.

Would you like further details, or do you have any questions?

---

Here are five related questions to consider:

1. How would the model change if the initial area were different?
2. What would the model look like if the area doubled every 6 hours instead?
3. How can we determine when the oil spill reaches a certain area?
4. What if the area grows at a different exponential rate—how would that affect $k$?
5. How can we use this model to estimate the spill’s area after 24 hours?

Tip: To solve exponential growth problems, identify the initial value and how quickly the quantity grows or shrinks, then use that information to find the growth constant.