Question

A city bus system carries 4000 passengers a day throughout a large city. The cost to ride the bus is $1.50 per person. The owner realizes that 100 fewer people would ride the bus for each $0.25 increase in fare. What should the bus system charge to produce the largest possible revenue?

Answer 1

the fare that should be charged to produce the largest possible revenue is $5.75 per person.

Explanation:

To determine the fare that would produce the largest possible revenue for the bus system, we need to find the fare at which the product of the number of passengers and the fare is maximized.

Let's assume the fare is x dollars. According to the given information, the number of passengers (P) can be represented as:

P = 4000 - 100 * (x - 1.50) / 0.25

The revenue (R) generated by the bus system can be calculated by multiplying the number of passengers (P) by the fare (x):

R = P * x

R = (4000 - 100 * (x - 1.50) / 0.25) * x

Expanding and simplifying the equation, we have:

R = (4000 - 400 * (x - 1.50)) * x

R = (4000 - 400x + 600) * x

R = (4600 - 400x) * x

R = 4600x - 400x²

To find the maximum value of R, we can take the derivative of R with respect to x and set it equal to zero:

dR/dx = 4600 - 800x = 0

800x = 4600

x = 4600 / 800

x = 5.75