# The owner of a baseball team and local stadium commissioned a study that showed the demand by fans for stadium seats (per playing date) to be P = 22 - 0.2Q, where P is the average price of a ticket and Q represents the number of seats (expressed in thousands). The local stadium seats a maximum of 56,000 per game. Suppose the owner offers you 10% of the revenues. If you can only choose a uniform per-ticket price, what is the maximum amount you can earn per game? Assume that all seats and all games are the same in this problem.)

The maximum you can earn per game is \$60,500.

The price at which the seats should be priced has to be determined to maximize the revenue earned. If the price of each seat is kept at P, the revenue from the stadium is given by:

R = P*Q

Substituting P = 22 - 0.2Q, gives

R = P*((22 - P)/0.2)

= 22*P/0.2-P^2/0.2

= 110P - P^2/0.2

Differentiating R with respect to P gives R' = 110-2*P/0.2 = 110-10P

The value of R is maximized at the point where R' is equal to 0.

Solving R' = 0 for P gives (110 - 10P) = 0

110 = 10P

P = 11

When P = 11, the demand is Q = (22 - 11)/0.2 = 55 (in thousands).

As this is lower than the maximum capacity of the stadium, it is possible to accommodate 55000 people in the stadium.

The revenue in this case is 11*55000 = 605000

As the owner offers 10% of the revenue, the earning per game is \$60,500.

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