Question
A town has a population of 2, point, 3, 5, times, 10, to the power 42.35×10
4
and shrinks at a rate of 9.1% every year. Which equation represents the town’s population after 2 years?
Answer
Multiple Choice Answers
P, equals, left parenthesis, 2, point, 3, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, plus, 0, point, 0, 9, 1, right parenthesis, squaredP=(2.35×10
4
)(1+0.091)
2
P, equals, left parenthesis, 2, point, 3, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 0, point, 0, 9, right parenthesis, squaredP=(2.35×10
4
)(0.09)
2
P, equals, left parenthesis, 2, point, 3, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, minus, 0, point, 0, 9, 1, right parenthesis, left parenthesis, 1, minus, 0, point, 0, 9, 1, right parenthesis, left parenthesis, 1, minus, 0, point, 0, 9, 1, right parenthesis, left parenthesis, 1, minus, 0, point, 0, 9, 1, right parenthesisP=(2.35×10
4
)(1−0.091)(1−0.091)(1−0.091)(1−0.091)
P, equals, left parenthesis, 2, point, 3, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 0, point, 9, 0, 9, right parenthesis, squaredP=(2.35×10
4
)(0.909)
2
4
and shrinks at a rate of 9.1% every year. Which equation represents the town’s population after 2 years?
Answer
Multiple Choice Answers
P, equals, left parenthesis, 2, point, 3, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, plus, 0, point, 0, 9, 1, right parenthesis, squaredP=(2.35×10
4
)(1+0.091)
2
P, equals, left parenthesis, 2, point, 3, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 0, point, 0, 9, right parenthesis, squaredP=(2.35×10
4
)(0.09)
2
P, equals, left parenthesis, 2, point, 3, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, minus, 0, point, 0, 9, 1, right parenthesis, left parenthesis, 1, minus, 0, point, 0, 9, 1, right parenthesis, left parenthesis, 1, minus, 0, point, 0, 9, 1, right parenthesis, left parenthesis, 1, minus, 0, point, 0, 9, 1, right parenthesisP=(2.35×10
4
)(1−0.091)(1−0.091)(1−0.091)(1−0.091)
P, equals, left parenthesis, 2, point, 3, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 0, point, 9, 0, 9, right parenthesis, squaredP=(2.35×10
4
)(0.909)
2
Answers
GPT-4o mini
To determine the correct equation that represents the town's population after 2 years, we need to consider the percentage decrease in population.
The town's population is shrinking at a rate of 9.1% each year. This means that each year, the population retains \(100\% - 9.1\% = 90.9\%\) of its population from the previous year. This is equivalent to multiplying the population by \(0.909\) each year.
After 2 years, the population will be:
\[
P = \text{initial population} \times (0.909)^2
\]
Given the initial population of \(2.35 \times 10^4\), the equation becomes:
\[
P = (2.35 \times 10^4) \times (0.909)^2
\]
Now, let's examine the choices based on this reasoning:
1. **Option 1**: \( P = (2.35 \times 10^4)(1 + 0.091)^2 \) - Incorrect, as it suggests an increase.
2. **Option 2**: \( P = (2.35 \times 10^4)(0.09)^2 \) - Incorrect, does not reflect the 9.1% decrease correctly.
3. **Option 3**: \( P = (2.35 \times 10^4)(1 - 0.091)(1 - 0.091) \) - Correct, this follows the logic of retaining 90.9% of population each year.
4. **Option 4**: \( P = (2.35 \times 10^4)(0.909)^2 \) - Correct, as it also correctly represents the decrease over 2 years.
Between options 3 and 4, both are valid but option 4 is more straightforward and uses a direct percentage. Therefore, the best choice representing the population after 2 years is:
**\( P = (2.35 \times 10^4)(0.909)^2 \)**.
The town's population is shrinking at a rate of 9.1% each year. This means that each year, the population retains \(100\% - 9.1\% = 90.9\%\) of its population from the previous year. This is equivalent to multiplying the population by \(0.909\) each year.
After 2 years, the population will be:
\[
P = \text{initial population} \times (0.909)^2
\]
Given the initial population of \(2.35 \times 10^4\), the equation becomes:
\[
P = (2.35 \times 10^4) \times (0.909)^2
\]
Now, let's examine the choices based on this reasoning:
1. **Option 1**: \( P = (2.35 \times 10^4)(1 + 0.091)^2 \) - Incorrect, as it suggests an increase.
2. **Option 2**: \( P = (2.35 \times 10^4)(0.09)^2 \) - Incorrect, does not reflect the 9.1% decrease correctly.
3. **Option 3**: \( P = (2.35 \times 10^4)(1 - 0.091)(1 - 0.091) \) - Correct, this follows the logic of retaining 90.9% of population each year.
4. **Option 4**: \( P = (2.35 \times 10^4)(0.909)^2 \) - Correct, as it also correctly represents the decrease over 2 years.
Between options 3 and 4, both are valid but option 4 is more straightforward and uses a direct percentage. Therefore, the best choice representing the population after 2 years is:
**\( P = (2.35 \times 10^4)(0.909)^2 \)**.