A particular city had a population of 23000 in 1950 and a population of 28000 in 1970. Assuming that its population continues to grow exponentially at a constant​ rate, what population will it have in 2000​?

To solve this problem, we can use the formula for exponential growth:

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

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 since the initial population.

### Step 1: Find the growth rate \( r \)
We know the population at two different times:
- In 1950 (\( t = 0 \)): \( P_0 = 23000 \)
- In 1970 (\( t = 20 \)): \( P(20) = 28000 \)

Using the formula:

\[
P(20) = 23000 \cdot e^{20r}
\]

We can solve for \( r \) by isolating it:

\[
28000 = 23000 \cdot e^{20r}
\]

\[
\frac{28000}{23000} = e^{20r}
\]

\[
\ln\left(\frac{28000}{23000}\right) = 20r
\]

\[
r = \frac{1}{20} \cdot \ln\left(\frac{28000}{23000}\right)
\]

### Step 2: Calculate the population in 2000
Once we have \( r \), we can predict the population in 2000 (\( t = 50 \)) using:

\[
P(50) = 23000 \cdot e^{50r}
\]

Let's calculate the exact values for \( r \) and \( P(50) \).The population of the city in the year 2000 is approximately **37,610**.

Would you like further details or have any questions?

Here are 5 related questions that expand on this topic:
1. How would the population change if the growth rate were higher or lower?
2. What would the population be in 1980 using the same growth model?
3. How does exponential growth differ from linear growth in population models?
4. What assumptions are being made in using the exponential growth model for population prediction?
5. How would you calculate the time it would take for the population to double?

**Tip:** Exponential growth models assume a constant percentage growth rate over time, which may not always accurately reflect real-world population changes, especially over long periods.

Explore how to predict population growth using exponential growth formula. Solve a problem involving a city's population growth from 1950 to 2000, with detailed steps and calculations. Suitable for advanced high school students and beyond.

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