A particular city had a population of 28 comma 000 in 1920 and a population of 31 comma 000 in 1930. Assuming that its population continues to grow exponentially at a constant​ rate, what population will it have in 2000​?
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Part 1
The population of the city in 2000 will be 
  
enter your response here people.
​(Round the final answer to the nearest whole number as needed. Round all intermediate values to six decimal places as​ needed.)

To solve this problem, we'll use the exponential growth model. The formula for exponential growth is:

\[
P(t) = P_0 \times e^{rt}
\]

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

Given dataTo solve this problem, we'll use the exponential growth model. The formula for exponential growth is:

\[
P(t) = P_0 \times e^{rt}
\]

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

Given 
- \( P_0 = 28,000 \) in 1920,
- \( P(10) = 31,000 \) in 1930.

First, we need to find the growth rate \( r \).

1. **Calculate the growth rate \( r \):**

\[
31,000 = 28,000 \times e^{10r}
\]

\[
\frac{31,000}{28,000} = e^{10r}
\]

\[
\frac{31}{28} = e^{10r}
\]

Take the natural logarithm of both sides:

\[
\ln\left(\frac{31}{28}\right) = 10r
\]

\[
r = \frac{\ln\left(\frac{31}{28}\right)}{10}
\]

Now, let's calculate \( r \):

\[
r \approx \frac{\ln(1.107142857)}{10} \approx \frac{0.10183}{10} \approx 0.010183
\]

2. **Calculate the population in 2000** (which is 80 years after 1920):

\[
P(80) = 28,000 \times e^{0.010183 \times 80}
\]

Calculate the exponent:

\[
0.010183 \times 80 \approx 0.81464
\]

Now, calculate the population:

\[
P(80) = 28,000 \times e^{0.81464} \approx 28,000 \times 2.258322 \approx 63,233
\]

So, the population of the city in the year 2000 will be approximately **63,233** people.

Would you like to go over the details or have any questions? 

Here are five related questions to consider:
1. How would the result change if the population in 1930 were different?
2. Can you derive the same result using a different growth model?
3. What is the population doubling time for this city based on the growth rate \( r \)?
4. How does exponential growth differ from linear growth in terms of population modeling?
5. How would the model change if the growth rate \( r \) varied over time?

**Tip:** Always round intermediate calculations carefully when working with exponential models, as small errors can significantly affect the final result.

A particular city had a population of 28,000 in 1920 and a population of 31,000 in 1930. Assuming that its population continues to grow exponentially at a constant​ rate, what population will it have in 2000​?

Calculate the population of a city in the year 2000 using the exponential growth model. Given the populations in 1920 and 1930, learn how to determine the growth rate and predict future population sizes. This problem involves exponential functions, logarithms, and population modeling, suitable for students in Grades 10-12.

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