Question
A town has a population of 68,000 and grows at a rate of 3.5% every year. Which equation represents the town’s population after 4 years?
Answer
Multiple Choice Answers
P, equals, 68, comma, 000, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesis, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesis, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesis, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesisP=68,000(1+0.035)(1+0.035)(1+0.035)(1+0.035)
P, equals, 68, comma, 000, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesisP=68,000(1+0.035)
P, equals, 68, comma, 000, left parenthesis, 1, minus, 0, point, 0, 3, 5, right parenthesis, to the power 4P=68,000(1−0.035)
4
P, equals, 68, comma, 000, left parenthesis, 0, point, 9, 6, 5, right parenthesis, to the power 4P=68,000(0.965)
4
Answer
Multiple Choice Answers
P, equals, 68, comma, 000, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesis, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesis, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesis, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesisP=68,000(1+0.035)(1+0.035)(1+0.035)(1+0.035)
P, equals, 68, comma, 000, left parenthesis, 1, plus, 0, point, 0, 3, 5, right parenthesisP=68,000(1+0.035)
P, equals, 68, comma, 000, left parenthesis, 1, minus, 0, point, 0, 3, 5, right parenthesis, to the power 4P=68,000(1−0.035)
4
P, equals, 68, comma, 000, left parenthesis, 0, point, 9, 6, 5, right parenthesis, to the power 4P=68,000(0.965)
4
Answers
GPT-4o mini
To determine the correct equation that represents the town's population after 4 years with an annual growth rate of 3.5%, we can use the formula for exponential growth:
\[ P = P_0(1 + r)^t \]
where:
- \( P \) is the future population,
- \( P_0 \) is the initial population,
- \( r \) is the growth rate,
- \( t \) is the number of years.
In this case:
- \( P_0 = 68,000 \)
- \( r = 0.035 \) (which is 3.5% expressed as a decimal)
- \( t = 4 \)
Plugging these values into the formula, we have:
\[
P = 68,000(1 + 0.035)^4
\]
Thus, this can also be expressed as:
\[
P = 68,000(1.035)^4
\]
Among the provided options, the first option accurately represents the population calculation over the 4 years:
**Option 1:**
\[ P = 68,000(1 + 0.035)(1 + 0.035)(1 + 0.035)(1 + 0.035) \]
This simplifies to the same as our exponential growth formula, and thus is the correct answer.
\[ P = P_0(1 + r)^t \]
where:
- \( P \) is the future population,
- \( P_0 \) is the initial population,
- \( r \) is the growth rate,
- \( t \) is the number of years.
In this case:
- \( P_0 = 68,000 \)
- \( r = 0.035 \) (which is 3.5% expressed as a decimal)
- \( t = 4 \)
Plugging these values into the formula, we have:
\[
P = 68,000(1 + 0.035)^4
\]
Thus, this can also be expressed as:
\[
P = 68,000(1.035)^4
\]
Among the provided options, the first option accurately represents the population calculation over the 4 years:
**Option 1:**
\[ P = 68,000(1 + 0.035)(1 + 0.035)(1 + 0.035)(1 + 0.035) \]
This simplifies to the same as our exponential growth formula, and thus is the correct answer.