Chapter 5.6, Problem 22PE

(a)

To determine

The objective function representing his weekly income for tutoring $x$ hrs. of chemistry and $y$ hours of math, if Guyton makes $\$24/\text{hr}$ tutoring chemistry and $\$20/\text{hr}$ tutoring math where $x$ represents the number of hours per week he spends tutoring chemistry and $y$ represents the number of hours per week he spends tutoring math.

(b)

To determine

The system of inequalities representing the constraints for the number hours spent tutoring each subject cannot be negative, he tutors at most $18\text{ hrs}$ per week, the tutoring center requires that he tutors math at least 4 hr per week and the demand for math tutors is greater than the demand for chemistry tutors so the number of hours he spends tutoring math must be at least twice the number of hours he spends tutoring chemistry if Guyton makes $\$24/\text{hr}$ tutoring chemistry and $\$20/\text{hr}$ tutoring math where $x$ represents the number of hours per week he spends tutoring chemistry and $y$ represents the number of hours per week he spends tutoring math.

(c)

To determine

To graph: The system of inequalities represented by the constraints $x\ge 0$, $y\ge 0$, $x+y\le 18$, $y\ge 4$, $y\ge 2x$.

(d)

To determine

To calculate: The vertices of the feasible region having bounding lines $x\ge 0$, $y\ge 0$, $x+y\le 18$, $y\ge 4$, $y\ge 2x$.

(e)

To determine

To calculate: The test of the objective function $z=24x+20y$ at each vertex $\left(0,4\right),\left(0,18\right),\left(6,12\right),\left(2,4\right)$.

(f)

To determine

The number hours Guytons should work tutoring math and chemistry to produce maximum profit.

(g)

To determine

To calculate: The maximize income of Guyton for objective function $z=24x+20y$.

(h)

To determine

Guyton maximize income is found at a point on the line $x+y=18$.