Question

Write an exponential function to model the following situation.
A population of 130,000 grows 5​% per year for 14 years.
How much will the population be after 14 ​years?

Answer 1

To model the population growth using an exponential function, we can use the formula:

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

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

Given:
- \( P_0 = 130,000 \) (initial population),
- \( r = 0.05 \) (5% growth rate),
- \( t = 14 \) years.

Now, we can plug these values into the formula:

\[
P(14) = 130,000 \cdot (1 + 0.05)^{14}
\]

Calculating \( (1 + 0.05)^{14} \):

\[
1 + 0.05 = 1.05
\]

Now, calculate \( 1.05^{14} \):

\[
1.05^{14} \approx 1.979
\]

Now, plug this back into our equation:

\[
P(14) \approx 130,000 \cdot 1.979
\]

Calculating:

\[
P(14) \approx 257,270
\]

Therefore, the population after 14 years will be approximately **257,270**.