To approach this problem, we start by understanding that we need to establish a linear relationship between the cost of tuition and the number of years since 1990. We'll identify the linear function [tex]\(C(x)\)[/tex] where [tex]\(x\)[/tex] is the number of years since 1990.
### Step-by-Step Solution:
1. Identify Given Information:
- Tuition in 1990, [tex]\(C(0)\)[/tex]: \[tex]$92
- Tuition in 2003, \(C(13)\): \$[/tex]287
Here, [tex]\(13\)[/tex] represents the number of years from 1990 to 2003.
2. Calculate the Slope (Rate of Change):
The slope [tex]\(m\)[/tex] of the linear function can be calculated as the rate of change in tuition costs per year. This is done using the formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given values:
[tex]\[
m = \frac{287 - 92}{13 - 0} = \frac{195}{13} = 15
\][/tex]
Thus, the slope [tex]\(m\)[/tex] is 15.
3. Establish the Linear Function:
The general form of a linear function is:
[tex]\[
C(x) = mx + b
\][/tex]
Where [tex]\(m\)[/tex] is the slope, and [tex]\(b\)[/tex] is the y-intercept (initial value when [tex]\(x = 0\)[/tex]).
Given that the tuition in 1990 ([tex]\(x = 0\)[/tex]) was \[tex]$92, \(b = 92\). Thus, the linear function \(C(x)\) is:
\[
C(x) = 15x + 92
\]
4. Calculate Tuition in 2010:
For the year 2010, \(x\) is the number of years since 1990:
\[
x = 2010 - 1990 = 20
\]
Substituting \(x = 20\) into the linear function:
\[
C(20) = 15 \cdot 20 + 92 = 300 + 92 = 392
\]
Therefore, the tuition in 2010 will be \$[/tex]392 per credit hour.
5. Determine the Year When Tuition Will Be \[tex]$527:
We need to find the year \(x\) when \(C(x) = 527\):
\[
527 = 15x + 92
\]
Solving for \(x\):
\[
527 - 92 = 15x
\]
\[
435 = 15x
\]
\[
x = \frac{435}{15} = 29
\]
The number of years since 1990 when the tuition will be \$[/tex]527 is 29 years. Adding this to 1990 gives:
[tex]\[
1990 + 29 = 2019
\][/tex]
Hence, the year when tuition will be \[tex]$527 per credit hour is 2019.
### Final Answers:
- The linear function \(C(x) = 15x + 92\)
- In the year 2010, tuition will be \$[/tex]392 per credit hour.
- In the year 2019, tuition will be \$527 per credit hour.
