Consider a (simplified) model of the town of Econville: all workers live in the residential district of town and must travel to the business district each day for work. Suppose there is currently a hill between the residential and business districts, so the road between the districts follows a rather indirect route. There are 6000 commuters in Econville and the existing road is large enough to accommodate them without congestion. A single trip from the residential district to the business district takes 30 minutes. To reduce travel times the local government considers building a tunnel through the hill. However, engineering constraints limit the size of the tunnel, so it may be congested. If x cars use the tunnel, it will take each of them 10 + x 200 minutes to get to the business district.

(f) (Challenging) Suppose x drivers take the tunnel each morning, meaning that 6000 − x use the existing road. Write down a formula for the total amount of time spent by drivers during their morning commutes. (Hint: repeat your working from (d), replacing the appropriate numbers with algebraic expressions involving x.)

(g) Draw a sketch of the formula from (f). Feel free to use a graphing calculator or a web tool such as Desmos (link). You may have to zoom out to see the graph.

(h) What number of drivers taking the tunnel would minimize the total amount of driving time each morning?

(i) Why is the privately-optimal outcome not the socially optimal one?

(j) How might the local government ensure only this many cars use the tunnel? If they achieve this, roughly how much time will this tunnel have saved after a year?