A particular city had a population of 25000 in 1930 and a population of 30000 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'll follow the same steps as before, using the exponential growth formula:

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

### Step 1: Find the growth rate \( r \)
Given:
- In 1930 (\( t = 0 \)): \( P_0 = 25000 \)
- In 1970 (\( t = 40 \)): \( P(40) = 30000 \)

Using the formula:

\[
30000 = 25000 \cdot e^{40r}
\]

We can solve for \( r \):

\[
\frac{30000}{25000} = e^{40r}
\]

\[
\ln\left(\frac{30000}{25000}\right) = 40r
\]

\[
r = \frac{1}{40} \cdot \ln\left(\frac{30000}{25000}\right)
\]

### Step 2: Calculate the population in 2000
To find the population in 2000 (\( t = 70 \)):

\[
P(70) = 25000 \cdot e^{70r}
\]

Let's calculate the values for \( r \) and \( P(70) \).The population of the city in the year 2000 is approximately **34,396**.

Would you like further details or have any questions?

Here are 5 related questions that expand on this topic:
1. How would the population in 2000 change if the growth rate was calculated using a different time interval?
2. What would the population be in 1980 using the same exponential growth model?
3. How does this population growth compare to the first city's growth rate?
4. What factors could cause deviations from this predicted population in real life?
5. How can exponential growth be applied to predict other phenomena, like financial investments?

**Tip:** When working with exponential growth, small changes in the growth rate or time period can lead to significant differences in the results.

Understand how to calculate population growth using exponential growth formula. Solve a problem where a city's population in 1930 and 1970 leads to predicting its population in 2000. Detailed steps and calculations provided.

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