Question

What is the exponential equation that models the population growth of a city if its initial population is 50,000 and it doubles every 20 years?

Answer 1

To model the population growth of a city using an exponential equation, we can use the formula for exponential growth, which is given by:

\[
P(t) = P_0 \cdot e^{kt}
\]

Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the growth constant,
- \( t \) is time in years,
- \( e \) is the base of the natural logarithm.

In this case, we know:
- The initial population \( P_0 = 50,000 \).
- The population doubles every 20 years.

To find \( k \), we can use the information that the population doubles in 20 years.

When the population doubles:
\[
P(20) = 2 \cdot P_0 = 2 \cdot 50,000 = 100,000
\]

Using the exponential growth formula:
\[
100,000 = 50,000 \cdot e^{20k}
\]

Dividing both sides by 50,000 gives:
\[
2 = e^{20k}
\]

Now, taking the natural logarithm of both sides:
\[
\ln(2) = 20k
\]

Solving for \( k \):
\[
k = \frac{\ln(2)}{20}
\]

Now we can write the exponential equation for the population growth:
\[
P(t) = 50,000 \cdot e^{\left(\frac{\ln(2)}{20}\right) t}
\]

This can also be expressed using \( 2^{t/20} \) because \( e^{\ln(2)} = 2 \):
\[
P(t) = 50,000 \cdot 2^{\frac{t}{20}}
\]

So, the exponential equation that models the population growth of the city is:

\[
P(t) = 50,000 \cdot 2^{\frac{t}{20}}
\]