A commuter train carries 600 passengers each day from a suburb to a city. It now costs $1 per person to ride the train. A study shows that 50 additional people will ride the train for each 5 cent reduction in fare. What fare should be charged in order to maximize total revenue?
A) 78 cents
B) 79 cents
C) 80 cents
D) 85 cents

Question

Quizplus · Accepted Answer

To maximize total revenue, we need to find the fare that maximizes the product of the number of passengers and the fare. Let x be the number of 5-cent reductions. The fare becomes (1 - 0.05x) dollars, and the number of passengers becomes (600 + 50x). The revenue R is given by R = (1 - 0.05x)(600 + 50x). Expanding and simplifying gives R = 600 + 50x - 30x - 2.5x^2. To find the maximum, take the derivative and set it to zero: dR/dx = 50 - 30 - 5x = 0, solving gives x = 4. The fare is then 1 - 0.05*4 = 0.80 or 80 cents.

A Commuter Train Carries 600 Passengers Each Day from a Suburb