Question

An athletic stadium holds 105,000 fans. With a ticket price of $22, the average attendance has been 32,000. When the price dropped to $16, the average attendance rose to 50,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue? Round ticket price to the nearest ten cents.

Answer 1

The revenue maximizing price will be $16.33.

Step-by-step explanation:

The capacity of an athletic stadium is 105,000 people.

When the ticket price is $22, the attendance is 32,000.

When the ticket price is $16, the attendance is 50,000.

The slope of the demand curve will be

=
`(50,000\ -\ 32,000 )/(16\ -\ 22)`

=
`(18,000)/(-6)`

= -3,000

Q = -3,000p + b

At p = 16, Q = 50,000

50,000 = -3,000 (16) + b

b = 98,000

The linear equation is

Q = -3,000p + 98,000

The total revenue will be

=
`Price\ *\ Quantity`

=
`-3,000p^2 + 98,000p`

The marginal revenue will be

=
`(d)/(dp) (TR)`

=
`(d)/(dp) (-3,000p^2 + 98,000p)`

= -6,000p + 98,000

The total revenue will be maximized when the marginal revenue is equal to zero.

-6,000p + 98,000 = 0

p =
`(98,000)/(6,000)`

p = 16.33