#### Answer

a. $z(x,y)=15x+10y$
b. $\begin{cases} x+y\leq20 \\ x\geq3\\ x\leq8 \end{cases}$
c. See graph
d. $z(3,0)= 45$, $z(8,0)= 120$, $z(3,17)= 215$, $z(8,12)= 240$
e. tutoring $8$ hours and working as a teacher's aid $12$ hours per week, maximum earning $240$ dollars.

#### Work Step by Step

a. Based on the given conditions, the total weekly earnings can be written as $z(x,y)=15x+10y$
b. We can convert the constraints into inequalities as
$\begin{cases} x+y\leq20 \\ x\geq3\\ x\leq8 \end{cases}$
c. See graph for the vertices and the solution region as a trapezoid shaped area.
d. With the objective equation and vertices, we have
$z(3,0)=15(3)+10(0)=45$, $z(8,0)=15(8)+10(0)=120$, $z(3,17)=15(3)+10(17)=215$, $z(8,12)=15(8)+10(12)=240$
e. Based on the above results, we can complete the missing portions as: tutoring $8$ hours per week and working as a teacher's aid for $12$ hours per week. The maximum amount that the student can earn each week is $240$ dollars.