what fare should we charge?

Question

GO transit offers a round trip ticket from Hamilton to Toronto for $20. Currently it averages a ridership of 10000 people per day, and has a maximum capacity of 15000. Mark research shows that for every $1 increase in price, 400 fewer people will ride the bus. What fare should GO transit charge to maximize revenue?

Answer 1

Let number of riders = R and Let the price = P. Let revenue = S.

dR/dP=-400 riders/dollar . Integrate to find R=-400P+C. Evaluate C by noting that R=10,000 with P=$20 finding C=18000 thus R=18000-400P. Sales = R*P = 18000P-400P². To find the maximum of S find dS/dP and set =0 which yields P=$22.50. Note that d²S/dP²= -800 <0 indicates that a local maximum occurs at P=22.5