The Randolph family likes nice dinners, but they are not crazy about preparing meals (or washing dishes). Their prandial utility function is U(h,t,r)=20h0.1t0.3r0.4 where h,t and r are the number of home-cooked dinners, takeout dinners and restaurant dinners, respectively, that the Randolphs consume in a month. The (average) prices of these dinners are ph=$10 (for home-cooked), pt=$25 (for takeout) and pr=$50 (for restaurant). a. (2 pts) Write down the Lagrangian for the Randolph's utility maximization problem given that their monthly dinner budget is B=$800. b. (5pts) Find the numbers of home-cooked, take-out and restaurant dinners that the Randolphs should consume per month to maximize their prandial utility (given their budget constraint), and find their maximum utility. c. (3 pts) Assuming that the prices and the utility function do not change, by approximately how much will the Randolphs need to increase their monthly dinner budget if they want to increase their prandial utility by 1 util (from the value you found in b.)