Data on annual tuition and average mid-career salary of graduates of a certain degree program are collected from a number of universities and colleges. The result of the data collection is the linear model $$ P_{n}=0.87 n+83,000 $$ where $$ n $$ annual tuition and $$ P_{n} $$ is the average mid-career salary of graduates, both in thousands of dollars. This model is accurate for annual tuition values between 10 thousand and 40 thousand dollars. (a) What is the rate of growth for this linear model? (b) According to this model, what is the average salary for a graduate of a college or university where the annual tuition is $$ \$ 25,000 $$ ? $$ \$ $$ (c) According to this model, what is the average salary for a graduate of a college or university where the annual tuition is $$ \$ 40,000 $$ ? $$ \$ $$

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### Given Information:
The linear model is given by the equation:
$$
P_n = 0.87n + 83,000
$$
where:
- $$ n $$ is the annual tuition in thousands of dollars.
- $$ P_n $$ is the average mid-career salary in thousands of dollars.

### (a) Rate of Growth
The rate of growth in a linear model is represented by the coefficient of $$ n $$. In this case, the coefficient of $$ n $$ is $$ 0.87 $$.

This means that for every additional thousand dollars in tuition, the average mid-career salary increases by $$ 0.87 $$ thousand dollars, or $$ \$ 870 $$.

### (b) Average Salary for $$ n = 25 $$ (Annual Tuition of $$ \$ 25,000 $$)
To find the average salary for a graduate where the annual tuition is $$ \$ 25,000 $$, we substitute $$ n = 25 $$ into the model:

$$
P_{25} = 0.87(25) + 83,000
$$

Calculating this step-by-step:
1. Calculate $$ 0.87 \times 25 $$:
   $$
   0.87 \times 25 = 21.75
   $$
2. Add $$ 83,000 $$:
   $$
   P_{25} = 21.75 + 83,000 = 83,021.75
   $$

Thus, the average salary for a graduate where the annual tuition is $$ \$ 25,000 $$ is approximately $$ \$ 83,021.75 $$.

### (c) Average Salary for $$ n = 40 $$ (Annual Tuition of $$ \$ 40,000 $$)
To find the average salary for a graduate where the annual tuition is $$ \$ 40,000 $$, we substitute $$ n = 40 $$ into the model:

$$
P_{40} = 0.87(40) + 83,000
$$

Calculating this step-by-step:
1. Calculate $$ 0.87 \times 40 $$:
   $$
   0.87 \times 40 = 34.8
   $$
2. Add $$ 83,000 $$:
   $$
   P_{40} = 34.8 + 83,000 = 83,034.8
   $$

Thus, the average salary for a graduate where the annual tuition is $$ \$ 40,000 $$ is approximately $$ \$ 83,034.80 $$.

### Summary of Results:
- (a) Rate of growth: $$ \$ 870 $$ per thousand dollars of tuition.
- (b) Average salary for $$ \$ 25,000 $$ tuition: $$ \$ 83,021.75 $$.
- (c) Average salary for $$ \$ 40,000 $$ tuition: $$ \$ 83,034.80 $$.
- (a) The model grows by \$870 for every \$1,000 increase in tuition. - (b) For \$25,000 tuition, the average salary is \$83,021.75. - (c) For \$40,000 tuition, the average salary is \$83,034.80.

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