Question 3: Suppose Duncan only engages in three activities; sleeping, playing Xbox, and working at Starbucks. He has to sleep 8 hours per night and the rest of his time he splits between playing Xbox (X hours per day) and working at Starbucks (S hours per day). He eats ramen noodles (It packages per day) which cost 2$ per package. His preferences over playing Xbox and eating ramen noodles are given by the utility function
U(X, R) = 5X3R
1. Write down Duncan's constrained optimization problem that determines how many hours he will play Xbox (X) and how many packages of ramen noodles he will eat (R).
2. Solve Duncan's problem if he earns 5$ an hour at Starbucks.
3. Now, suppose Duncan gets 20$ a day from his parents. Solve Duncan's new problem.