Question

A town has a population of 1, point, 2, 3, times, 10, to the power 41.23×10
4
and grows at a rate of 6.7% every year. Which equation represents the town’s population after 4 years?
Answer
Multiple Choice Answers
P, equals, left parenthesis, 1, point, 2, 3, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, point, 0, 6, 7, right parenthesis, to the power 4P=(1.23×10
4
)(1.067)
4

P, equals, left parenthesis, 1, point, 2, 3, times, 10, to the power 4 , right parenthesis, left parenthesis, 0, point, 0, 6, 7, right parenthesis, to the power 4P=(1.23×10
4
)(0.067)
4

P, equals, left parenthesis, 1, point, 2, 3, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, minus, 0, point, 0, 6, 7, right parenthesis, to the power 4P=(1.23×10
4
)(1−0.067)
4

P, equals, left parenthesis, 1, point, 2, 3, times, 10, to the power 4 , right parenthesis, left parenthesis, 1, point, 6, 7, right parenthesis, to the power 4P=(1.23×10
4
)(1.67)
4

Answer 1

To represent the town's population after 4 years, we can use the formula for exponential growth due to a constant growth rate, which is given by:

\[
P = P_0 \times (1 + r)^t
\]

where:
- \(P_0\) is the initial population,
- \(r\) is the growth rate (as a decimal),
- \(t\) is the time in years.

In this case:
- The initial population \(P_0 = 1.23 \times 10^4\),
- The growth rate \(r = 6.7\% = 0.067\),
- The time \(t = 4\).

Therefore, our population equation after 4 years would be:

\[
P = (1.23 \times 10^4) \times (1 + 0.067)^4
\]

This can be simplified to:

\[
P = (1.23 \times 10^4) \times (1.067)^4
\]

Now, looking at the multiple-choice options, the correct representation of the town’s population after 4 years is:

**\(P = (1.23 \times 10^4)(1.067)^4\)**

Thus, the answer is:

\[
\text{P = } (1.23 \times 10^4)(1.067)^4
\]