When first observed, an oil spill covers 2 square miles. Measurements show that the area is tripling every 6 hrs. Find an exponential model for the area A (in mi2) of the oil spill as a function of time t (in hr) from the beginning of the spill.
A(t)

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.

When first observed, an oil spill covers 2 square miles. Measurements show that the area is tripling every 6 hrs. Find an exponential model for the area A (in mi2) of the oil spill as a function of time t (in hr) from the beginning of the spill.

Discover how to derive an exponential model for the growth of an oil spill, where the area triples every 6 hours. Learn step-by-step how to use exponential growth formulas and logarithms to model the spill's area over time. This problem is ideal for students studying exponential functions in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation