Question

Mr. Carr derives a lot of pleasure from driving under the wide blue skies. For the number of miles x that he drives, he receives Utility U(x) = 500x – x2 Once he drives beyond a certain number of miles weariness kicks in and the ride becomes less and less enjoyable. His car gives him a decent highway mileage of 25 miles per gallon. But paying for gas (or paying in general) induces disutility shown by U(y) = -1000y, where y is the number of $ spent. Mr. Carr is willing to spend at most $25 for leisurely driving each week. (general hint on the following: the decision variable will always be just the number of miles drive) a) Find the optimum number of miles driven each week by Mr. Carr given that the price of gas is $2.50 per gallon? b) How does that value change if the price of gas rises to $5.00 per gallon? c) Now, further assume that there is a probability of 0.001 of Mr. Carr getting a flat tire for every mile he drives. The disutility he derives from a flat tire is U(z) = -50000z, where z is the number of flat tires incurred, and each flat tire cost $50 to replace. Find the distance driven by Mr. Carr that maximizes his utility taking into account the expected likelihood of flat tires (assume again that the price of gas is $2.50 per gallon).

Answer 1

4) u(x) = 500x-x^ using unconstrained mapimization Copremization of one variable case. 4CX) = 500 - 2x > 500-2x = 0 (condition -or) => x=250. (critical value) u"(x) = -2 20 , maximum. ( sufficient condition) . Given the price of gas as 25/ gallom, the opteinal no. of miles doiven each weak is 250 miles. this can be proved too! Mileage of the car = 25 miles per gallon i gf needs 10 gallons to run 250miles. He is ready to spend at most $25. Given price an 2.5 gallon TE - 2.5X10 - 95-y. I Amount he needs to run 250 miles. ( $2.5x 10 galloc) b) of the price of gas rises to $5)- gallon Then, Te to run 250 miles will be $5x10 = 504, but he is willing to spend $25 at most. . when price will be $5/ gallon. The total mile travelled will be 125 miler. Prove : Mileage - 25 miles per gallon - It need s gallons to run 125 :: TE=35x5 = 25$=y milen

(CI Əhe probability of getting a frat tire = 0.001 . Given price of gas = $2.5/ gallon . He will havel 250 miles. .. Elgetting feat fire) = Plgetting frat hire) x no of miles travelled. = 250x 0.001 = 0.25. It means that if he travels a50 miles then the expected fear fire úill occur iso-as, which cs insignificant as it : 1. is less than a.s. by Mr. Carr will maximise hej ulici driving 250 miles given the price of gas ($2.5), the proba. -bility of having flat tire (0.001) and income he is willing to spend at most ($25).