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The area of undeveloped land in a suburban town is decreasing at a rate of [tex]$17.3 \%$[/tex] annually. In 2016, there were 3,400 acres of undeveloped land. If [tex]$t$[/tex] 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?

A. [tex]$900 = 3,400(0.827)^t$[/tex]
B. [tex]$3,400 = 900(0.173)^t$[/tex]
C. [tex]$900 = 3,400(1.173)^t$[/tex]
D. [tex]$3,400 = 900(0.9827)^t$[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to understand the concept of exponential decay. The area of undeveloped land decreases at a constant percentage rate annually, which we can model using the formula for exponential decay:

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

where:
- [tex]\( A_0 \)[/tex] is the initial amount of undeveloped land,
- [tex]\( A \)[/tex] is the remaining amount of undeveloped land after time [tex]\( t \)[/tex],
- [tex]\( r \)[/tex] is the annual decay rate,
- [tex]\( t \)[/tex] is the time in years.

In this scenario:
- The initial amount of undeveloped land [tex]\( A_0 \)[/tex] is 3400 acres.
- The remaining amount of undeveloped land [tex]\( A \)[/tex] is 900 acres.
- The annual decay rate [tex]\( r \)[/tex] is 17.3%, which can be represented as a decimal: [tex]\( 0.173 \)[/tex].

Substituting these values into the formula, we get:

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

Simplifying the expression inside the parentheses:

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

This matches equation A:

[tex]\[ 900 = 3400(0.827)^t \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{A} \][/tex]

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