Question

A university is trying to determine what price to charge for tickets to football games. At a price of $27 per ticket, attendance averages 40,000 people per game. Every decrease of $3 adds 10,000 people to the average number.

Every person at the game spends an average of $3.00 on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price?

Answer 1

the university should charge $13.50 per ticket to maximize revenue, and 85,000 people will attend at that price.

Explanation:

To find the price per ticket that maximizes revenue, we need to find the price that maximizes the product of the number of tickets sold and the revenue per ticket. Revenue per ticket is equal to the ticket price plus the average concession spending per person, or $27 + $3 = $30.

Let x be the number of $3 decreases in ticket price, so the ticket price can be expressed as $27 - $3x. The number of people attending can be expressed as 40,000 + 10,000x.

Therefore, the revenue can be expressed as:

Revenue = (Ticket price) x (Number of tickets sold) x (Concession spending per person)

Revenue = ($27 - $3x) x (40,000 + 10,000x) x $3.00

Expanding this expression, we get:

Revenue = $81,000,000 + $270,000,000x - $30,000,000x^2

To maximize revenue, we need to find the value of x that maximizes this quadratic function. We can do this by finding the vertex of the parabola, which is located at:

x = -b / 2a

where a = -30,000,000, b = 270,000,000, and c = 81,000,000

x = -270,000,000 / 2(-30,000,000)

x = 4.5

Since x represents the number of $3 decreases in ticket price, we can find the optimal ticket price by subtracting 3 times x from the original price of $27:

Optimal ticket price = $27 - $3(4.5) = $13.50

The optimal number of people attending can be found by substituting x = 4.5 into the expression for the number of people attending:

Number of people attending = 40,000 + 10,000(4.5) = 85,000