Ten thousand people commute every day to a large urban center. These commuters can take the train, which has capacity for all 10,000 people and takes 70.00 minutes, or they can drive. Driving takes 35.00 minutes with no traffic, but driving is often subject to congestion. When others are driving, driving time is represented by Td=35.00+(N/100), where N is the number of commuters who have chosen to drive.

Assume that commuters want to get to work as quickly as possible and will thus choose to drive or take the train based solely on which will allow them to arrive sooner. How many commuters will choose to drive?

How many minutes will it take them to get to the city?

How many total person minutes (TPM) will be spent commuting each day? (Total person minutes are simply the time spent commuting for each commuter times the number of commuters.)

A city planner is bothered by the inefficiency of all the time spent commuting. He would like to minimize the total person minutes spent commuting each day. How many people should drive to minimize the total person minutes spent commuting?

How many minutes will it take to commute by car if that many people drive?

The city planner is pleased to have determined the number of drivers that will minimize TPM and thus increase productive work hours for many people. However, he realizes that his solution needs some form of incentive to prevent too many people from driving. The planner thus decides to charge a toll. If the typical worker values her time at $0.40 per minute, how large does the toll need to be to result in the efficient number of people driving?