https://file.aiquickdraw.com/mathbot/file/1728512371266x1ir820g.jpg

Let's break down the problem step by step based on the image:

### Part 1
1. **Slope Calculation**: The graph represents the population of a military base over several years. To calculate the slope of the line (rate of population growth), we can use the formula for slope:
   
   \[
   \text{slope} = \frac{\Delta y}{\Delta x} = \frac{\text{change in population}}{\text{change in years}}
   \]

   Using two points from the graph:
   - At year 0, the population is 865.
   - At year 10, the population is 1045.

   So, the slope is:

   \[
   \text{slope} = \frac{1045 - 865}{10 - 0} = \frac{180}{10} = 18 \, \text{people per year}
   \]

2. **Initial Value**: The initial value (when \( n = 0 \)) is the population at the year 0 on the graph, which is 865 people.

### Part 2
1. **Formula for Population**: The population \( P \) is modeled as a linear function of the years since 1980, denoted by \( n \). The equation for a line is:

   \[
   P = \text{slope} \cdot n + \text{initial value}
   \]

   Substituting the slope and initial value:

   \[
   P = 18n + 865
   \]

2. **Predict Population in 1993**: The year 1993 is 13 years after 1980, so \( n = 13 \). Using the formula \( P = 18n + 865 \):

   \[
   P = 18 \cdot 13 + 865 = 234 + 865 = 1099 \, \text{people}
   \]

It seems there might be a discrepancy in the answers you're trying to input. If you're using different points or different interpretations for slope or initial value, that could affect the result.

Would you like further clarification or to discuss any part of this?

### Related Questions:
1. What is the interpretation of the slope in terms of population growth?
2. How does changing the initial value affect the formula for population?
3. If the slope were steeper, what would that mean for the population growth?
4. Can you calculate the population for a future year, like 2025, using this model?
5. How would the formula change if the base’s population decreased over time?

**Tip**: Always ensure you're choosing points on the graph that are clearly labeled when calculating slope, as small misinterpretations can lead to significant differences in the result.

Calculate the slope and initial value from the graph of a military base population, and find a formula to predict population growth based on years since 1980.

Learn how to calculate the slope of a line representing the population growth of a military base, determine the initial population, and create a formula to predict future population based on the years since 1980. This problem involves key algebraic concepts like slope-intercept form and is suitable for students in Grades 9-11.

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