The population P of a city is given by
P = 58,500ekt
where t represents the year, with 
t = 0
 corresponding to 2000. In 2002, the population was 62,000. Find the value of k. (Round your answer to five decimal places.)
k = 


Use this result to predict the population in 2020. (Round your answer to the nearest whole number.)
 people

We are given the equation for population growth:

\[
P = 58,500 e^{kt}
\]

where \( t = 0 \) corresponds to the year 2000, and we need to find the value of \( k \). We know that in the year 2002, the population was 62,000, which means when \( t = 2 \), \( P = 62,000 \).

### Step 1: Solve for \( k \)

Substitute the known values into the equation for \( P \):

\[
62,000 = 58,500 e^{k(2)}
\]

Divide both sides by 58,500 to isolate \( e^{2k} \):

\[
\frac{62,000}{58,500} = e^{2k}
\]

\[
1.05983 = e^{2k}
\]

Now take the natural logarithm of both sides to solve for \( 2k \):

\[
\ln(1.05983) = 2k
\]

\[
0.05813 = 2k
\]

Now divide by 2 to find \( k \):

\[
k = \frac{0.05813}{2} = 0.02907
\]

Thus, the value of \( k \) is approximately:

\[
k \approx 0.02907
\]

### Step 2: Predict the population in 2020

To predict the population in 2020, we use \( t = 20 \) (since \( t = 0 \) corresponds to the year 2000). Substituting \( t = 20 \) and \( k = 0.02907 \) into the equation:

\[
P = 58,500 e^{0.02907(20)}
\]

Simplify the exponent:

\[
P = 58,500 e^{0.5814}
\]

Now calculate \( e^{0.5814} \):

\[
P = 58,500 \times 1.78841
\]

\[
P \approx 104,683
\]

Thus, the predicted population in 2020 is approximately:

\[
P \approx 104,683 \text{ people}
\]

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Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How does changing the value of \( k \) affect the population growth rate?
2. What would the population be in 2030 using the same model?
3. How do exponential growth models compare to linear growth models in predicting long-term populations?
4. If the population growth rate slows down, how would this affect the value of \( k \)?
5. How can we adjust the model if external factors like migration significantly influence the population?

**Tip:** Always check the units and meaning of the constants in population growth models to ensure correct interpretation.

The population P of a city is given by P = 58,500e^kt where t represents the year, with t = 0 corresponding to 2000. In 2002, the population was 62,000. Find the value of k. Use this result to predict the population in 2020.

Learn how to find the growth rate constant k in an exponential population model and predict the population in 2020. This problem involves solving for k using logarithms and exponential growth equations. Suitable for Grades 10-12.

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