Chapter 8.6, Problem 22PE

Guyton makes $\text{\$}24\text{/hr}$ tutoring chemistry and $\text{\$}20\text{/hr}$ tutoring math.

Let $x$ represent the number of hours per week he spends tutoring chemistry. Let $y$ represent the number of hours per week he spends tutoring math.

a. Write an objective function representing his weekly income for tutoring $x$ hours of chemistry and $y$ hours of math.

b. The time that Guyton devotes to tutoring is limited by the following constraints. Write a system of inequalities representing the constraints.

• The number of hours spent tutoring each subject cannot be negative.

• Due to the academic demands of his own classes he tutors at most $18\text{ hr}$ per week.

• The tutoring center requires that he tutors math at least $4\text{ hr}$ per week.

• The demand for math tutors is greater than the demand for chemistry tutors. Therefore, the number of hours he spends tutoring math must be at least twice the number of hours he spends tutoring chemistry.

c. Graph the system of inequalities represented by the constraints.

d. Find the vertices of the feasible region.

e. Test the objective function at each vertex

f. How many hours tutoring math and how many hours tutoring chemistry should Guyton work to maximize his income?

g. What is the maximum income?

h. Explain why Guyton's maximum income is found at a point on the line $x+y=18.$