May you explain it step by step, how the answer was derived and how the utility was arrived, coefficient of the utility function?

Sylvester and Rachel live together on an island with two goods, food and shelter. Food is a private good and shelter is a public good. The utility function of Sylvester is U = F; + 2(Zs + 2,.) where F5 is the amount of food he buys, ZS is the amount of shelter he buys and Z, is the amount of shelter Rachel buys. Similarly, the utility function of Rachel is U = 15;. + ZS + 2,. where E. is the amount of food she buys. Suppose the price of food is 10 and both Sylvester and Rachel have 100 to spend. Suppose also that it is not possible for Sylvester and Rachel to reach any kind of agreement. Determine the quantity of shelter they will buy and the efcient quantity of shelter if the price of shelter is 5? What is the quantity of shelter they will buy if the price of the shelter is 15? Both individuals consider food and shelter to be substitutes based on their preferences. This means that their marginal rate of substitution between food and shelter is constant along their indifference curve. NH}: 1 For Rachel: MRSR = MUs = I= 1 For Sylvester: MRSS = 144? _ 1 S For perfect substitutes we expect there to be a corner solution to the optimization problem. To solve the optimization problem we compare the marginal rate of substitution with the price ratio. Both Sylvester and Rachel will spend all of their income on shelter when the price of shelter is 5. This gives Sylvester a utility of 80 and Rachel a utility of40. Total utility is 120 and this is eicient. When the price of shelter is 15, Rachel will spend all of her income of food and Sylvester will spend all of his income on shelter. This generates a total utility of 30. This is not e'icient, because had they spent all of their income on shelter at these prices then total utility would be 40