Question

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

Answer 1

Answer:

Ticket price 11,5 $  

R(max)  =  132750 $

Step-by-step explanation:INCOMPLETE QUESTION

The ticket price is missing

We will assume 12 $

Let call x the decreasing number of the ticket price

The revenue is

R =  ticket price * numbers of spectator

R(x) =  (  12 - x ) *  ( 11000 + 1000*x )   for all   0 ≤ x  ≤ 4   ( the seating capacity is 15000 )

R(x) =  132000 + 12000*x  - 11000*x - 1000*x²

R(x) =  132000 + 1000*x  - 1000*x²

Taking derivatives on both sides of the equatio we get:

R´(x)  =  1000  -  2000x

R´(x)  = 0    ⇒    1000  -  2000x  =  0

x  =  1000/2000    ⇒  x = 0,5 $

We notice that R´´(x)  < 0    then we have a maximum for  x = 0,5

and the price is 12 - 0,5  = 11,5 $ for ticket

and the R(max) is

R(max) = 132000 + 1000*0,5  + 1000* (0,5)²

R(max) = 132000 + 500 + 250

R(max) = 132750 $