Auniversity is trying to determine what price to charge for tickets to football games. at a price of $18 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 $4.50 on concessions. what price per ticket should be charged in order to maximize revenue? how many people will attend at that price?

They should charge $12.75 per ticket.  If they do, 57,500 people will attend.

We start the equation with (18-3x).  We don't know how many times we're subtracting $3 from the price.  This is multiplied by (40,000+10,000x).  We will add 10,000 people the same number of times that we subtract $3 from the price.  This represents the amount of money for tickets multiplied by the amount of people.  We still need to add the amount of concessions.  Since (40,000+10,000x) represents the number of people, we multiply that by $4.50, the average amount each person spends on concessions.  This gives us:

We set it equal to 0 because when we are solving a quadratic equation, the solutions are where it intersects the x-axis, where y=0.

We multiply our terms on both pieces next:


When we find the maximum of a quadratic, we must find the axis of symmetry.  That is done using x=-b/2a:



This means the new price will be 18-3*1.75=12.75.

The new attendance will be 40,000+10,000*1.75=57,500.

How many groups are there?