Question
A town has a population of 2, point, 2, 5, times, 10, to the power 42.25×10
4
and grows at a rate of 5.1% every year. Which equation represents the town’s population after 2 years?
Answer
Attempt 1 out of 3
Multiple Choice Answers
P, equals, left parenthesis, 2, point, 2, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 0, point, 0, 5, 1, right parenthesis, squaredP=(2.25×10
4
)(0.051)
2
P, equals, left parenthesis, 2, point, 2, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, minus, 0, point, 0, 5, 1, right parenthesis, squaredP=(2.25×10
4
)(1−0.051)
2
P, equals, left parenthesis, 2, point, 2, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, plus, 0, point, 0, 5, 1, right parenthesis, left parenthesis, 1, plus, 0, point, 0, 5, 1, right parenthesisP=(2.25×10
4
)(1+0.051)(1+0.051)
P, equals, left parenthesis, 2, point, 2, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, plus, 0, point, 0, 5, 1, right parenthesisP=(2.25×10
4
)(1+0.051)
4
and grows at a rate of 5.1% every year. Which equation represents the town’s population after 2 years?
Answer
Attempt 1 out of 3
Multiple Choice Answers
P, equals, left parenthesis, 2, point, 2, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 0, point, 0, 5, 1, right parenthesis, squaredP=(2.25×10
4
)(0.051)
2
P, equals, left parenthesis, 2, point, 2, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, minus, 0, point, 0, 5, 1, right parenthesis, squaredP=(2.25×10
4
)(1−0.051)
2
P, equals, left parenthesis, 2, point, 2, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, plus, 0, point, 0, 5, 1, right parenthesis, left parenthesis, 1, plus, 0, point, 0, 5, 1, right parenthesisP=(2.25×10
4
)(1+0.051)(1+0.051)
P, equals, left parenthesis, 2, point, 2, 5, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, plus, 0, point, 0, 5, 1, right parenthesisP=(2.25×10
4
)(1+0.051)
Answers
Answered by
GPT-4o mini
To find the population of a town growing at a rate of 5.1% per year after 2 years, you can use the formula for exponential growth:
\[
P = P_0(1 + r)^t
\]
Where:
- \( P \) is the future population.
- \( P_0 \) is the current population.
- \( r \) is the growth rate (expressed as a decimal).
- \( t \) is the number of years.
In this case:
- The initial population \( P_0 = 2.25 \times 10^4 \).
- The growth rate \( r = 5.1\% = 0.051 \).
- The time \( t = 2 \) years.
Therefore, the equation that represents the town’s population after 2 years is:
\[
P = (2.25 \times 10^4)(1 + 0.051)^2
\]
Now let's look at the multiple-choice answers to find the one that matches:
1. **\( P = (2.25 \times 10^4)(0.051)^2 \)** - Incorrect. This does not represent the growth properly.
2. **\( P = (2.25 \times 10^4)(1 - 0.051)^2 \)** - Incorrect. This represents a decrease in population.
3. **\( P = (2.25 \times 10^4)(1 + 0.051)(1 + 0.051) \)** - Correct. This represents the population after two years of growth.
4. **\( P = (2.25 \times 10^4)(1 + 0.051) \)** - Incorrect. This represents only one year of growth.
Thus, the correct answer is:
**\( P = (2.25 \times 10^4)(1 + 0.051)(1 + 0.051) \)**.
\[
P = P_0(1 + r)^t
\]
Where:
- \( P \) is the future population.
- \( P_0 \) is the current population.
- \( r \) is the growth rate (expressed as a decimal).
- \( t \) is the number of years.
In this case:
- The initial population \( P_0 = 2.25 \times 10^4 \).
- The growth rate \( r = 5.1\% = 0.051 \).
- The time \( t = 2 \) years.
Therefore, the equation that represents the town’s population after 2 years is:
\[
P = (2.25 \times 10^4)(1 + 0.051)^2
\]
Now let's look at the multiple-choice answers to find the one that matches:
1. **\( P = (2.25 \times 10^4)(0.051)^2 \)** - Incorrect. This does not represent the growth properly.
2. **\( P = (2.25 \times 10^4)(1 - 0.051)^2 \)** - Incorrect. This represents a decrease in population.
3. **\( P = (2.25 \times 10^4)(1 + 0.051)(1 + 0.051) \)** - Correct. This represents the population after two years of growth.
4. **\( P = (2.25 \times 10^4)(1 + 0.051) \)** - Incorrect. This represents only one year of growth.
Thus, the correct answer is:
**\( P = (2.25 \times 10^4)(1 + 0.051)(1 + 0.051) \)**.