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The problem asks for an equation that models the decay of the area of coral reefs over time and to use that equation to find how many years it will take for a reef to decay from 350 km² to 108 km², given a 3.5% decay rate per year.

### Step-by-Step Solution:

1. **Exponential Decay Formula**:
   The formula for exponential decay is:
   \[
   A(t) = A_0 \cdot e^{-kt}
   \]
   Where:
   - \( A(t) \) is the area after \( t \) years.
   - \( A_0 \) is the initial area (in this case, 350 km²).
   - \( k \) is the decay constant, which can be calculated using the given decay rate.
   - \( t \) is the time in years.
   
2. **Decay Rate**:
   The coral reef is decaying at a rate of 3.5% per year. So the decay rate \( k \) can be calculated as:
   \[
   k = \frac{3.5}{100} = 0.035
   \]

3. **Equation**:
   The equation for the area of the coral reef after \( t \) years is:
   \[
   A(t) = 350 \cdot e^{-0.035t}
   \]

4. **Finding Time for 108 km²**:
   Now, we need to find \( t \) when the area \( A(t) = 108 \) km². Using the equation:
   \[
   108 = 350 \cdot e^{-0.035t}
   \]
   Solving for \( t \):
   \[
   \frac{108}{350} = e^{-0.035t}
   \]
   \[
   \ln\left(\frac{108}{350}\right) = -0.035t
   \]
   \[
   t = \frac{\ln\left(\frac{108}{350}\right)}{-0.035}
   \]
   Now, calculating:
   \[
   t = \frac{\ln(0.30857)}{-0.035} \approx \frac{-1.175}{-0.035} \approx 33.57
   \]
   Rounding to the nearest year, it will take about **34 years** for the reef to decay to 108 km².

### Summary:
- **Equation**: \( A(t) = 350 \cdot e^{-0.035t} \)
- **Time**: It will take approximately **34 years** for the coral reef to decay to an area of 108 km².

Would you like a more detailed explanation on any part? Here are some related questions:

1. What are the general properties of exponential decay?
2. How do you calculate the decay constant \( k \) for different rates?
3. What is the significance of the natural logarithm in solving exponential decay problems?
4. How can this model be applied to other real-world scenarios?
5. Can this method be used to calculate coral reef growth if it were instead increasing in size?

**Tip**: Always check units in exponential decay problems to ensure consistency, especially when working with percentages and time intervals.

Coral reefs throughout the world are dying at a rate of about 3.5% per year. Write an equation that can be used to determine the future area of a reef that now has an area of 350 km². Let c be the area of the coral reef in t years from now. Use the equation to determine how many years until the reef decays to an area of 108 km². Round to the nearest year.

Explore how to use exponential decay equations to model the decrease in coral reef area over time. Given a 3.5% annual decay rate, determine the number of years it will take for a reef to shrink from 350 km² to 108 km².

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