Q$8(12$ points$)$

Darja is trying to optimize her one free day per week by choosing to either spend her hours doing house tasks $($which give her $5$ units of satisfaction per hour$)$ or watching trashy tv $($which give her $4$ units of satisfaction per hour$).$ Each of these tasks makes her feel relaxation, bur also uses energy. For each hour she spends doing house tasks, she feels $2$ units of relaxation and uses $12$ units of energy. For each hour she spends watching trashy tv she feels $8$ units of relaxation and uses $3$ units of energy. She would like to feel at least $50$ units of relaxation in her day, but use no more that $100$ units of energy. If she has has only $8$ total hours in her free day, how many hours of each type of activity should she do in order to maximize her units of satisfaction?

SET UP THE PROBLEM AS ASKED BELOW, BUT DO NOT SOLVE.

a$)$ Define your variables.

b$)$ Write the objective function for this problem.

c$)$ Write out your constraints.