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The question involves coral reefs declining at a rate of 1.5% per year, and you're asked to write an equation, determine its family (exponential), and find how many years it will take for a reef's area to decrease from 300 km² to 188 km².

Let's break it down:

### Part (a): Writing the equation
This is an exponential decay problem, and the formula for exponential decay is generally:

\[
c(t) = c_0 \times (1 - r)^t
\]

Where:
- \( c(t) \) is the area of the reef after \( t \) years,
- \( c_0 = 300 \, \text{km}^2 \) is the initial area,
- \( r = 0.015 \) is the annual decay rate (1.5%),
- \( t \) is the time in years.

So, the equation becomes:

\[
c(t) = 300 \times (1 - 0.015)^t = 300 \times (0.985)^t
\]

### Part (b): The equation family
Since the equation involves exponential decay, the family of the equation is **exponential**.

### Part (c): Calculating the time to reach 188 km²
We want to find \( t \) when the area \( c(t) = 188 \, \text{km}^2 \). Set up the equation and solve for \( t \):

\[
188 = 300 \times (0.985)^t
\]

To solve for \( t \), divide both sides by 300:

\[
\frac{188}{300} = (0.985)^t
\]

\[
0.6267 = (0.985)^t
\]

Now, take the natural logarithm (ln) of both sides to solve for \( t \):

\[
\ln(0.6267) = t \times \ln(0.985)
\]

\[
t = \frac{\ln(0.6267)}{\ln(0.985)}
\]

Using a calculator:

\[
t = \frac{-0.4671}{-0.0151} \approx 30.93
\]

So, it will take approximately **31 years** for the reef's area to decay to 188 km².

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Do you need any more details or have any questions?

### Related Questions:
1. What is the formula for exponential growth, and how does it differ from decay?
2. What other factors could contribute to the decay of coral reefs besides the rate given?
3. How would the equation change if the decay rate was 2% instead of 1.5%?
4. What is the significance of using natural logarithms in solving exponential equations?
5. Can exponential decay ever reach zero? Why or why not?

### Tip:
When solving exponential equations, always isolate the exponential term before taking logarithms for easier computation.

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 300 km². Let c be the area of the coral reef in t years from now. Use this equation to find how many years it will take for the area to decay to 188 km².

Learn how to model the exponential decay of coral reefs, where the area decreases by 1.5% per year. This problem explores exponential decay equations, solving for time, and natural logarithms to determine how long it takes for a reef's area to shrink from 300 km² to 188 km².

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