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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. Which equation can be used to determine after how many years the town will have 900 acres of undeveloped land remaining?

A. [tex]\(3,400 = 900(0.9827)^t\)[/tex]

B. [tex]\(900 = 3,400(1.173)^t\)[/tex]

C. [tex]\(900 = 3,400(0.827)^t\)[/tex]

D. [tex]\(3,400 = 900(0.173)^t\)[/tex]

Sagot :

GinnyAnswer
To determine which equation correctly represents the situation, let's break down the given information and identify the correct relationship:

1. The area of undeveloped land is decreasing at a rate of 17.3% annually. This means that each year, 17.3% of the land is lost. Conversely, 82.7% of the land remains each year (since 100% - 17.3% = 82.7%, or 0.827 as a decimal).

2. We start with 3,400 acres in 2016. We want to find after how many years ([tex]\( t \)[/tex]) the land will be reduced to 900 acres.

3. The relationship between the initial amount of land ([tex]\( A_0 \)[/tex]), the remaining amount ([tex]\( A \)[/tex]), the decay factor (0.827), and time ([tex]\( t \)[/tex]) can be expressed as:
[tex]\[ A = A_0 \times (0.827)^t \][/tex]

4. Plugging in the initial amount ([tex]\( A_0 = 3400 \)[/tex] acres) and the remaining amount ([tex]\( A = 900 \)[/tex] acres), the equation becomes:
[tex]\[ 900 = 3400 \times (0.827)^t \][/tex]

Thus, this equation correctly represents the relationship and the time it takes for the undeveloped land to decrease to 900 acres in terms of the initial 3,400 acres at the given decay rate.

Reviewing the choices:

A. [tex]\(3400=900(0.9827)^t\)[/tex] - The initial and remaining amounts are incorrect, and the decay factor is also not accurate.
B. [tex]\(900=3400(1.173)^t\)[/tex] - The decay factor is incorrect (it should be less than 1, not greater than 1).
C. [tex]\(900=3400(0.827)^t\)[/tex] - Correct equation, as shown in the steps above.
D. [tex]\(3400=900(0.173)^t\)[/tex] - The initial and remaining amounts are swapped, and the decay factor is incorrect.

The correct answer is:

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