A university is trying to determine what price to charge for tickets to football games. At a price of $\mathrm{\$}28$ per ticket, attendance averages 40,000 people per game. Every decrease of $\mathrm{\$}4$ adds 10,000 people to the average number. Every person at the game spends an average of $\mathrm{\$}6.00$ on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price?
What is the price per ticket?
What is the average attendance?
people

Step 1 ：A university is trying to determine what price to charge for tickets to football games. At a price of $28perticket,attendanceaverages40,000peoplepergame.Everydecreaseof$4 adds 10,000 people to the average number. Every person at the game spends an average of $6.00 on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price?

Step 2 ：The revenue from ticket sales is the product of the price per ticket and the number of tickets sold. The revenue from concessions is the product of the number of people attending and the average amount spent on concessions. The total revenue is the sum of these two amounts. We need to find the price per ticket that maximizes this total revenue.

Step 3 ：We can express the number of people attending as a function of the price per ticket. If we let $p$ be the price per ticket, then the number of people attending is $40000+10000\ast (28-p)/4$. This is because for every $4 decrease in price, 10000 more people attend.

Step 4 ：The revenue from ticket sales is then $p\ast (40000+10000\ast (28-p)/4)$ and the revenue from concessions is $6\ast (40000+10000\ast (28-p)/4)$. The total revenue is the sum of these two amounts.

Step 5 ：After calculating the total revenue for different ticket prices, we find that the maximum revenue is $1560000.0whentheticketpriceis$20.

Step 6 ：Final Answer: The price per ticket that should be charged in order to maximize revenue is $\boxed{{\displaystyle 20}}$ dollars.