Question

A hockey team plays in an arena with a seating capacity of 15, 000. At $12 per ticket, they have an average attendance of 11, 000 fans. A recent survey shows that for every dollar they decrease their ticket price, their attendance will increase by 1, 000 fans. Where should they set their price for maximum revenue?

Answer 1

$11.50

Explanation:

Let R = Revenue function = Price of ticket × booked seats

Let n = number of times lowered the price of ticket by $1.

Price = ($12 - n × $1) = 12 - n dollars

Quantity = number of sold tickets (1100) + n (1000) = 11,000 + 1,000n spectators

Therefore, R(n) = (12-n) (11000 + 1000n) = 132,000 + 1000n - 1000n²

= -1000 (x² - x - 132)

= -1000 ((x - 1/2)² - 529/4)

= -1000 (x - 1/2)² + 132,250

R(n) - 13,250 = -1,000 (x - 1/2)²

n = 0.5 ⇒ 12 - 0.5 = $11.50

Spectators = 11,000 + 1,000 (1/2) = 11,500

Revenue = $13,250

They should set their price at $11.50 for maximum revenue.