A hockey team plays in an arena that has a seating capacity

A hockey team plays in an arena that has a seating capacity of 15,000 spectators. With the ticket price set at $50. average attendance at recent games has been 10500. A market survey indicates that for each dollar the ticket price is lowered, the average attendance increases by 100. Find a function that models the revenue in terms of ticket price. Which attendance will give the maximum revenue? Justify your answer.

Solution

with average attendance of spectators -- 10500 priced at $50

for each dollar price decreased avge attendance increases by 100

Let the price get reduced by $x so, new price 50 -x and average attendance increases by 10500 +100x

a) So, Revenue function = (50 -x)(10500 +100x)

b) Maximum Revenue is found at vertex of the quadratic equation : 525000 -5500x -100x^2

Vertex x= -b/2a = -(-5500)/(2*-100) = -27.5

So, new price = 50 - (-27.5) = $77.5

Avg Attendance = 10500 + 100(-27.5) = 7750

So, we can see that maximum revenue is attained when price is increased by $ 27.5 to $ 77.5 and avg attendance is 7750

So, revenue = $ 600625