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The problem involves modeling the decay of coral reefs over time using an exponential decay equation. Let's break it down:

### a. The equation provided is:
\[
c(t) = 300(1 - 0.015)^t
\]
Where:
- \( c(t) \) is the area of the coral reef in square kilometers after \( t \) years.
- 300 km² is the initial area of the coral reef.
- 1.5% decay rate per year, meaning 0.015 represents the percentage loss per year.
- \( t \) is the number of years.

### b. The equation is from the **exponential decay** family since it models a quantity that decreases by a fixed percentage per time period.

### c. To find how many years until the area decays to 188 km², we can use the equation and solve for \( t \):
\[
188 = 300(1 - 0.015)^t
\]
We now solve for \( t \).

1. Divide both sides by 300:
\[
\frac{188}{300} = (1 - 0.015)^t
\]
\[
0.6267 = (0.985)^t
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(0.6267) = t \ln(0.985)
\]

3. Solve for \( t \):
\[
t = \frac{\ln(0.6267)}{\ln(0.985)}
\]
Using a calculator:
\[
t \approx \frac{-0.4677}{-0.01511} \approx 30.96
\]
Rounding to the nearest year, \( t \approx 31 \) years.

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

Would you like further details or have any questions?

### Related Questions:
1. What are the key differences between exponential growth and decay?
2. How would the equation change if the decay rate were 2% instead of 1.5%?
3. What would be the area of the reef after 10 years?
4. How do we interpret the constant 0.985 in the decay equation?
5. How would the decay equation change if the initial area was 400 km² instead of 300 km²?

### Tip:
When solving exponential equations, always isolate the exponential term first before applying logarithms to solve for the variable.

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.

Explore how to model the decay of coral reefs using an exponential decay equation with a rate of 1.5% per year. Calculate how many years it will take for the reef to shrink from 300 km² to 188 km², using logarithms to solve for time.

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