(Solved): 5. (18 pts) The city of String Valley is squeezed between two mountains and is 36 miles long, runn ...

5. (18 pts) The city of String Valley is squeezed between two mountains and is 36 miles long, running from north to south, and only about 1 block wide. Within the town, the population has a uniform density of 100 people per mile. Because of the rocky terrain, nobody lives outside the city limits on either the north or the south edge of town. Because of strict zoning regulations, the city has only three bowling alleys that are identical. One of these is located at the city limits on the north edge of town, one of them is located at the city limits on the south edge of town, and one is located at the exact center of town. Travel costs including time and gasoline are $\$1$ per mile. All of the citizens of the town have the same preferences. They are willing to bowl once a week if the cost of bowling including travel costs and the price charged by the bowling alley does not exceed $\$15$. (a) (2 pts) For questions (a) to (d), assume that these bowling alleys face no competition from the other bowling alleys in town. Consider one of the bowling alleys at either edge of town. If it charges $\$10$ for a night of bowling, how far will a citizen of String Valley be willing to travel to bowl there? How many customers would this individual bowling alley have per week? (b) (3 pts) Write a formula for the number of customers that a bowling alley at the edge of town will have if it charges Sp per night of bowling. Write a formula for this bowling alley's inverse demand function. (c) (3 pts) Suppose that the bowling alleys at the end of town have a marginal cost of $\$3$ per customer and set their prices to maximize profits. How many customers will they have? What price will they charge? How far away from the edge of town does their most distant customer live? (d) (4 pts) Now consider the bowling alley in the center of town. If it charges a price of $\$\mathrm{p}$, how many customers will it have per week? If it also has marginal costs of $\$3$ per customer and maximizes its profits, what price will it charge? How many customers will it have per week? How far away from the center of town will its most distant customers live? (e) (6 pts) Suppose that the city relaxes its zoning restrictions on where the bowling alleys can locate, but continues to issue operating licenses to only 3 bowling alleys. Both of the bowling alleys at the end of town are about to end their leases and can locate anywhere in town that they like at about the same cost. The bowling alley in the center of town is committed to stay where it is. Assume that all the bowling alleys have a marginal cost of $\$3$. Would either of the alleys at the edge of town improve its profits by locating next to the existing bowling alley in the center of town? Explain. What would be a profit-maximizing location for each of these two bowling alleys?