Question

a city had a population of 110,000 in 1920. the population grew at a rate of 1% per year thereafter. what exponential function could model this equation? use an F(x) type function. Use the function to estimate the population in 1922.

Answer 1

To model the population of the city with an exponential growth 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 (in decimal form),
- \(t\) is the number of years since the initial population.

Given:
- \(P_0 = 110,000\)
- \(r = 0.01\) (1% growth rate)
- \(t = 1920 - 1920 = 0\) for the year 1920

Thus, the function becomes:

\[
P(t) = 110000 \cdot (1 + 0.01)^t
\]

or

\[
P(t) = 110000 \cdot (1.01)^t
\]

Now, to estimate the population in 1922, we need to find \(t\) for the year 1922:

\[
t = 1922 - 1920 = 2
\]

Now we substitute \(t = 2\) into the function:

\[
P(2) = 110000 \cdot (1.01)^2
\]

Calculating \( (1.01)^2 \):

\[
(1.01)^2 = 1.0201
\]

Now, substituting this back into the equation:

\[
P(2) = 110000 \cdot 1.0201 \approx 112221
\]

Thus, the estimated population in 1922 is approximately **112,221**.