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Answer :
To determine which equation represents the town's population after 2 years given a population of 126,000 and a shrink rate of 4.3% per year, we need to use the concept of exponential decay.
1. Understand the decay factor:
- A 4.3% shrinkage means the population decreases by 4.3% each year.
- Convert the percentage to a decimal for calculations: [tex]\(4.3\% = 0.043\)[/tex].
- The decay factor for one year is calculated as [tex]\(1 - 0.043 = 0.957\)[/tex].
2. Apply the decay factor over 2 years:
- We need to apply the decay factor for each of the 2 years. This means we multiply the original population by the decay factor squared.
- So, after 2 years, the population will be:
[tex]\[
P = 126,000 \times (0.957)^2
\][/tex]
3. Evaluate the expression:
- Using this formula, the population after 2 years is approximately [tex]\(115,396.974\)[/tex].
Therefore, the equation that correctly represents the town's population after 2 years is:
[tex]\[
P = 126,000 \times (0.957)^2
\][/tex]
This matches the option [tex]\(P=126,000(0.957)(0.957)\)[/tex].
1. Understand the decay factor:
- A 4.3% shrinkage means the population decreases by 4.3% each year.
- Convert the percentage to a decimal for calculations: [tex]\(4.3\% = 0.043\)[/tex].
- The decay factor for one year is calculated as [tex]\(1 - 0.043 = 0.957\)[/tex].
2. Apply the decay factor over 2 years:
- We need to apply the decay factor for each of the 2 years. This means we multiply the original population by the decay factor squared.
- So, after 2 years, the population will be:
[tex]\[
P = 126,000 \times (0.957)^2
\][/tex]
3. Evaluate the expression:
- Using this formula, the population after 2 years is approximately [tex]\(115,396.974\)[/tex].
Therefore, the equation that correctly represents the town's population after 2 years is:
[tex]\[
P = 126,000 \times (0.957)^2
\][/tex]
This matches the option [tex]\(P=126,000(0.957)(0.957)\)[/tex].
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