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.) Now, his car gives him a decent highway mileage of 25 miles to the gallon. But paying for gas, represented by y, induces disutility for Mr. Carr, shown by U(y) = -1, 000y. Mr. Carr is willing to spend up to $25 for leisurely driving every week.
a. Find the optimum number of miles driven by Mr. Carr every week, given that the price of gas is $2.50 per gallon.
b. How does that value change when the price of gas rises to $5.00 per gallon?
c. Now, further assume that there is a probability of 0.001 that Mr. Carr will get a flat tire every mile he drives. The disutility from a flat tire is given by U(z) = -50,000z (where z is the number of flat tires incurred), and each flat tire costs $50 to replace. Find the distance driven that maximizes Mr. Carr's utility after taking into account the expected likelihood of flat tires (assume that the price of gas is $2.50 per gallon).