Question

a City's transit authority serves 178500 communters daily when the fare is $1.90. Market research has determined that every penny decrease in the fare will result in 1,050 new riders. what fare will maximize revenue?

Answer 1

revenue = #riders * fare
= (178500+1050(190-x))(.01x) for x<=190

that's just a parabola with vertex at x=180.

Answer 2

number of penny decreases ---- n

cost of fare = 190-n
number of riders = 178500 + 1050n

revenue = R = (190-n)(178500+1050n)
P' = (190-n)(1050) + (178500 + 1050n)(-1)
= 0 for a max of P

178500 + 1050n = 1050(190-n)
divide by 1050
170 + n = 190-n
2n = 20
n = 10

the fare should be 190+10 = 200
or it should be $ 2.00