Question

Write an exponential function to model the following situation.
A population of 140 comma 000 grows 3​% per year for 14 years.
How much will the popluation be after 14 ​years?

Answer 1

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

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

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

In your case:
- The initial population \(P_0 = 140,000\),
- The growth rate \(r = 3\% = 0.03\),
- The time \(t = 14\) years.

Substituting these values into the formula gives:

\[
P(14) = 140,000 \times (1 + 0.03)^{14}
\]

Now calculate \(P(14)\):

1. Calculate \(1 + 0.03 = 1.03\).
2. Raise \(1.03\) to the power of \(14\):

\[
1.03^{14} \approx 1.5107 \text{ (using a calculator)}
\]

3. Multiply this by the initial population:

\[
P(14) \approx 140,000 \times 1.5107 \approx 211,498
\]

Therefore, the estimated population after 14 years will be approximately **211,498**.

So the exponential function modeling this situation is:

\[
P(t) = 140,000 \times (1.03)^t
\]

And after 14 years, the population will be approximately **211,498**.