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The problem involves coral reefs decaying at a rate of 1.5% per year, with an initial area of 450 km². Let’s solve each part step by step.

### Part a: The equation
The equation modeling exponential decay can be written as:

\[
A(t) = A_0 \cdot e^{-kt}
\]

Where:
- \(A(t)\) is the area after \(t\) years,
- \(A_0\) is the initial area (450 km²),
- \(k\) is the decay constant,
- \(t\) is the number of years.

Since the rate of decay is 1.5% per year, we know:

\[
k = 0.015
\]

Thus, the equation becomes:

\[
A(t) = 450 \cdot e^{-0.015t}
\]

### Part b: Family of equation
This is an **exponential decay** equation.

### Part c: Determining how many years until the reef decays to 273 km²
We use the equation from part (a):

\[
273 = 450 \cdot e^{-0.015t}
\]

First, divide both sides by 450:

\[
\frac{273}{450} = e^{-0.015t}
\]

\[
0.6067 = e^{-0.015t}
\]

Next, take the natural logarithm of both sides:

\[
\ln(0.6067) = -0.015t
\]

\[
-0.4995 = -0.015t
\]

Now, solve for \(t\):

\[
t = \frac{-0.4995}{-0.015} \approx 33.3
\]

So, the reef will decay to 273 km² in about **33 years**.

Would you like further clarification on any part? Here are some additional questions you might find helpful:

1. What happens if the decay rate increases or decreases?
2. How would the equation change if the decay rate were compounded monthly instead of yearly?
3. Can you explain the difference between exponential growth and exponential decay?
4. What would the area be after 10 years?
5. How can we use this model to predict the remaining area at any given time?

**Tip:** Always check units when solving exponential decay problems to ensure consistency in the equation!

Coral reefs throughout the world are dying at a rate of about 1.5% per year. Write an equation that can be used to determine the future area of a reef that now has an area of 450 km². Let t be the years from now.

A) Write the equation.

B) Identify the type of equation.

C) Use the equation to determine how many years until the reef decays to an area of 273 km².

Solve a real-world problem involving exponential decay. Learn how to model coral reef decay with an exponential equation and use logarithms to calculate the time it takes for a reef to shrink from 450 km² to 273 km². Perfect for high school algebra students.

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