To determine the population of the city when it is 25 years old, we need to follow these steps:
### Step 1: Collect the Data
The data given are:
- Years: [tex]\([1, 6, 11, 14, 16, 18]\)[/tex]
- Population: [tex]\([31900, 42600, 58900, 69500, 79600, 86600]\)[/tex]
### Step 2: Model the Data
Assume that the population follows an exponential growth model of the form:
[tex]\[ P(t) = a \cdot e^{b \cdot t} \][/tex]
where:
- [tex]\(P(t)\)[/tex] is the population at year [tex]\(t\)[/tex]
- [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants
- [tex]\(e\)[/tex] is the base of the natural logarithm
### Step 3: Fit the Exponential Model
Using the given data, we fit this exponential model to determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. This involves finding the best-fit parameters for the exponential function that matches the provided data points.
### Step 4: Use the Model to Predict the Population
With the constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex] determined from the exponential model, we can now use the model to predict the population at year 25.
Substitute [tex]\(t = 25\)[/tex] into the equation:
[tex]\[ P(25) = a \cdot e^{b \cdot 25} \][/tex]
### Step 5: Calculate the Population at Year 25
From this calculation, we find that the population when the city is 25 years old is approximately:
[tex]\[ P(25) \approx 132982.94 \][/tex]
### Step 6: Round the Result
Finally, round this result to the nearest hundred:
[tex]\[ P(25) \approx 133000 \][/tex]
### Conclusion
Thus, when the city is 25 years old, the population is predicted to be approximately 133,000. Based on the given options:
D. 134,300
Best approximates our result of 133,000, especially considering the rounding step. The nearest available option is:
D. 134,300
