Let's solve the problem step-by-step to find the correct equation that represents the scenario given:
1. Understand the problem:
- We have an initial amount of undeveloped land: 3,400 acres in 2016.
- The land is decreasing at a rate of 17.3% per year.
- We need to find an equation that tells us after how many years the remaining undeveloped land will be 900 acres.
2. Identify the components of an exponential decay equation:
- The general form of an exponential decay equation is [tex]\( A = A_0(1 - r)^t \)[/tex], where:
- [tex]\( A \)[/tex] is the amount remaining after time [tex]\( t \)[/tex],
- [tex]\( A_0 \)[/tex] is the initial amount,
- [tex]\( r \)[/tex] is the decay rate (expressed as a decimal),
- [tex]\( t \)[/tex] is the time in years.
3. Substitute the given values:
- Initial amount, [tex]\( A_0 \)[/tex]: 3,400 acres.
- Remaining amount, [tex]\( A \)[/tex]: 900 acres.
- Decay rate, [tex]\( r \)[/tex]: 17.3% or 0.173.
4. Form the equation:
- Based on the exponential decay formula, we have:
[tex]\[
900 = 3400(1 - 0.173)^t
\][/tex]
- Simplifying inside the parentheses:
[tex]\[
1 - 0.173 = 0.827
\][/tex]
- Thus, the equation becomes:
[tex]\[
900 = 3400(0.827)^t
\][/tex]
5. Match with the provided options:
- Option A: [tex]\( 900 = 3,400(1.173)^t \)[/tex]
- Option B: [tex]\( 3,400 = 900 (0.173)^t \)[/tex]
- Option C: [tex]\( 900 = 3,400(0.827)^t \)[/tex]
- Option D: [tex]\( 3,400 = 900(0.9827)^t \)[/tex]
After comparing our derived equation with the options provided, we see that the correct equation is given by:
[tex]\[
\boxed{900 = 3400(0.827)^t}
\][/tex]
Thus, the correct answer is:
[tex]\[ C. \; 900 = 3,400(0.827)^t \][/tex]
