The area of undeveloped land in a suburban town is decreasing at a rate of 17.3% annually. In 2016, there were 3,400 acres of undeveloped land. If t represents the number of years since 2016, which equation can be used to determine after how many years the town will have 900 acres of undeveloped land remaining?

To model the problem, we can use the equation for exponential decay, which is given by:

\[
A(t) = A_0 \times (1 - r)^t
\]

Where:
- \( A(t) \) is the amount of undeveloped land at time \( t \) (in acres),
- \( A_0 \) is the initial amount of undeveloped land (in this case, 3,400 acres),
- \( r \) is the annual rate of decrease (17.3%, or 0.173 in decimal form),
- \( t \) is the number of years since 2016.

We are asked to find after how many years the undeveloped land will be 900 acres, so we want to solve for \( t \) when \( A(t) = 900 \).

Substitute the given values into the equation:

\[
900 = 3400 \times (1 - 0.173)^t
\]

Simplifying:

\[
900 = 3400 \times (0.827)^t
\]

To solve for \( t \), divide both sides by 3400:

\[
\frac{900}{3400} = (0.827)^t
\]

\[
0.2647 = (0.827)^t
\]

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

\[
\ln(0.2647) = t \times \ln(0.827)
\]

Solving for \( t \):

\[
t = \frac{\ln(0.2647)}{\ln(0.827)}
\]

Now you can use a calculator to compute \( t \). Would you like me to calculate it for you, or do you have any questions?

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Here are 5 relative questions:
1. What happens to the undeveloped land if the rate of decrease changes?
2. How would the equation change if the annual rate of decrease were 10% instead of 17.3%?
3. How long would it take for the land to decrease to 1,000 acres instead of 900 acres?
4. What is the significance of using a natural logarithm in solving exponential equations?
5. How can exponential decay be applied to population growth models?

**Tip:** When working with exponential equations, logarithms are essential for isolating the variable in the exponent.

Learn how to use exponential decay to model the reduction of undeveloped land in a suburban town. The problem involves solving for the number of years after 2016 until the undeveloped land decreases to 900 acres, using the formula A(t) = A_0 * (1 - r)^t. This step-by-step guide covers exponential decay and logarithmic functions, ideal for high school students in Grades 9-12.

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