Two food trucks are opening up in Harrisonburg: Hamilton's Hot Dogs and Smith's Sandwiches both food truck owners have paid off their debts and neither business has fixed costs. The two businesses produce differentiated products, Hamilton has marginal cost of $5, and Smith has marginal cost of $10. Inverse demand for the products are ph = 45-9h-1.5qs for Hamilton's Hot Dogs, Ps= 40 qs qh for Smith's Sandwiches, and the two food trucks engage in Cournot competition. (a) (4 points) Write down a function that gives Smith's Sandwiches total profits as a function of its own output and Hamilton's output. Your function should contain qħh, qs, and numerical constants only. (b) (6 points) Derive Hamilton's best-response function given the output of Smith's Sandwiches. Derive Smith's best-response function given the output of Hamilton's Hot Dogs. (c) (10 points) What are the Cournot-Nash equilibrium production levels for each firm? What are the profits earned for each firm? How large is consumer surplus for each product? Be careful calculating the vertical intercept for each demand curve. (d) (4 points) For this final part, assume the firms produce a homogeneous product, compete á la Bertrand, and demand is given by q = 45-min(Ph, Ps). Marginal costs remain as above. Provide one set of prices charged by the two firms that would constitute a Bertrand-Nash equilibrium. No math is necessary and there are multiple correct options. Explain your finding.