Please answer the question provided in the image below

Question l: This question asks you to reconsider the model of optimal sin taxes that we studied in class with a different distribution of types. Assume that everyone has p 65 and y 40. Assume further that proportion of the population has [3 0.85 while proportion I has [3 I (both types have (a) As a function of and t, what is the uniform lump-sum transfer? (b) As a function of and t, derive an expression for social welfare. (c) Solve for the optimal tax. (d) How does the optimal tax depend on Q? Provide some intuition for this answer.