A student has two sources of expenses: living and consumption. if the student lives in an apartment with s square meters and consumes c consumables then they receive utility

u (s, c) = s^1/5 c^4/5

^{The student chooses apartment size and consumption amount to get the highest utility. Total l living expenses are 1000+rs. Each unit of consumption costs 1. The students spends exactly 1800.}

^{Building restrictions require apartments to be at least 20 square meters but no more than 120 square meters.}

^{a. List all the cases that will occur from complementary slackness, with the implied equations and checks for each case.}

^{b. Under what conditions will the student live in a 20 m ^2 apartment? a 120M^2 apartment ? an apartment between 20 m^2 and 120 m^2?}