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A new college opened in 1998. It started with 500 students. By 2013, the college had 3125 students. Assume the growth of the student population can be modeled with a linear equation.

Be sure to show calculations supporting your answer.

a. What is the slope of the linear model for the number of students at the college?
b. What are the units for this slope?



Answer :

GinnyAnswer
Let's solve the problem step-by-step:

### Part (a): Calculating the Slope of the Linear Model

1. Determine the initial and final years.

- Initial year: 1998
- Final year: 2013

2. Determine the initial and final number of students.

- Initial number of students in 1998: 500
- Final number of students in 2013: 3125

3. Determine the number of years the college has been open.

The number of years the college has been open is the difference between the final and initial years:
[tex]\[ \text{Years open} = \text{Final year} - \text{Initial year} = 2013 - 1998 = 15 \text{ years} \][/tex]

4. Calculate the change in the number of students.

The change in the number of students is the difference between the final and initial number of students:
[tex]\[ \Delta \text{Students} = \text{Final number of students} - \text{Initial number of students} = 3125 - 500 = 2625 \text{ students} \][/tex]

5. Calculate the slope of the linear model.

The slope ([tex]\(m\)[/tex]) of the linear model can be calculated using the formula:
[tex]\[ \text{Slope} = \frac{\Delta \text{Students}}{\Delta \text{Years}} = \frac{2625 \text{ students}}{15 \text{ years}} = 175 \text{ students per year} \][/tex]

### Part (b): Determining the Units for the Slope

1. Identify the units for the numerator and denominator of the slope.

- Numerator (the change in the number of students): Students
- Denominator (the change in years): Years

2. Determine the units for the slope.

The units for the slope are obtained from the ratio of the change in the number of students to the change in years:
[tex]\[ \text{Units for the slope} = \frac{\text{Students}}{\text{Years}} = \text{Students per year} \][/tex]

### Conclusion

1. The slope of the linear model for the number of students at the college is 175.
2. The units for this slope are students per year.

Therefore, the college's student population increased by 175 students per year from 1998 to 2013.

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