Problema Solution
A city's transit authority serves 182,000 commuters daily when the fare is $1.60. Market research has determined that every penny decrease in the fare will result in 1,300 new riders. What fare will maximize revenue?
Answer provided by our tutors
Let x represent the number of one-penny ($0.01) decreases in the fare.
New price = (1.60 - 0.01x)
Number of riders = (182000 + 1300x)
Revenues = (New price)·(Number of riders)
Let R(x) represent the revenues then we have:
R(x) = (1.60 - 0.01x)(182000 + 1300x)
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click here to see the simplification of the function
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R(x) = - 13x^2 + 260x + 291200
Since the quotient in front of x^2 is - 13 < 0 the function has maximum for x max = -b/2a where a = -13, b = 260
x max = -260/(2*(-13))
x max = 10
For x = 10 the fare is: 1.60 - 0.01*10 = 1.60 - 0.10 = $1.50
The fare of $1.50 will maximize the revenue.