Question

A hockey team plays in an arena with a seating capacity of 15 000 spectators. With ticket prices set at $12, average attendance at a game has been 11 000. A market surveys indicates that for each dollar that ticket prices are lowered, the average attendence will increase by 1000. How should the owners of the team set ticket prices so as to maximize their revenue from ticket sales?

Answer 1

Answer:

their price at $11.50 for maximum revenue

Explanation:

given data

seating capacity = 15000

ticket prices = $12

average attendance game = 11000

average attendance increase = 1000

solution

we consider here Revenue function is R and that is express as

Revenue function R = Price of ticket × booked seats    .....................1

here number of time lowered the price of ticket n = $1

so here price will be as

price = $12 - ( n × $1)

price = $(12 - n)       ................2

so here Quantity will express as

Quantity = number of sold tickets + n (1000)

Quantity = 1100 + n (1000)   Spectators   ..................3

so R(n) will be

R(n) = 12-n (11000 + 1000n)

R(n)  = 132,000 + 1000n - 1000n²

R(n) = -1000 (x² - x - 132)

R(n) = -1000 ( (x - 0.5)² - $\frac{529}{4}$ )

so

R(n)  = -1000 (x - 0.5)² + 132,250

R(n) - 13,250 = -1,000 (x - 0.5)²

solve it we get

n = 0.5     ..........4

so from equation 2

price = 12 - 0.5

price = $11.50

so from equation 3

Spectators = 11,000 + 1,000 (0.5)

Spectators = 11,500

and

Revenue = $13,250

so that here set their price at $11.50 for maximum revenue