Could you please help me solve the below question:

4. (25 points) A student has two sources of expenses: living and consump- tion. If the student lives in an apartment with 5 square metres and con- sumes c consumables, then they receive utility U'IIP- u(s,c) : sic . The student chooses apartment size and consumption amount to get the highest utility. Total living expenses are 1000 + rs. Each unit of consump- tion costs 1. The student spends exactly 1800. a) (4 points) Building restrictions require apartments to be at least 20 square metres but no more than 120 square metres. Formally write the constrained optimisation problem. b) (16 points) List all cases that will occur from complementary slack- ness, with the implied equations and checks for each case. c) (3 points) Under what conditions will the student live in a 207712 apart- ment? A 1207112 apartment? An apartment between 207112 and 1207112? d) (2 point) Now suppose each unit of consumption costs 2 (instead of 1). Everything else stays the same. Without solving the problem again, use the envelope theorem to approximate change in utility