A baseball team plays in a stadium that holds 54,000 spectators. With the ticket price at $10, the average attendance at recent games has been 21,000. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 3000.
(a) Find a function that models the revenue in terms of ticket price. (Let x represent the price of a ticket and R represent the revenue.) R(x) = _________
(b) Find the price that maximizes revenue from ticket sales. $______________
(c) What ticket price is so high that no revenue is generated? $___________

Answer:

a) R(x) =  51000 x - 3000 x²

b) When the Revenue is maximum, The price is $ 8.5

c) When No revenue is generated, The price of the ticket is $17

Explanation:

Given - A baseball team plays in a stadium that holds 54,000 spectators. With the ticket price at $10, the average attendance at recent games has been 21,000. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 3000.

To find - (a) Find a function that models the revenue in terms of ticket price.  

              (b) Find the price that maximizes revenue from ticket sales.  

              (c) What ticket price is so high that no revenue is generated?

Proof -

Let us assume that,

The price of a ticket = x

The revenue = R

Now,

a)

Total number of tickets sold , N = 21000 + 3000(10 - x)

                                                     = 21000 + 30000 - 3000 x

                                                     = 51000 - 3000 x

⇒Total number of tickets sold , N = 51000 - 3000 x

Now,

Revenue , R(x) = x * N

                        = x [ 51000 - 3000 x ]

                        = 51000 x - 3000 x²

⇒R(x) =  51000 x - 3000 x²

b)

For maximum Revenue,

Put [tex]\frac{dR(x)}{dx} = 0[/tex]

Now,

[tex]\frac{dR(x)}{dx}[/tex] = 51000 - 6000 x

⇒51000 - 6000 x = 0

⇒6000 x = 51000

⇒6x = 51

⇒x = [tex]\frac{51}{6}[/tex] = 8.5

∴ we get

When the Revenue is maximum, The price = $ 8.5

c)

If No Revenue is generated

⇒R(x) = 0

⇒51000 x - 3000 x² = 0

⇒1000 x ( 51 - 3 x ) = 0

⇒1000 x = 0, 51 - 3 x = 0

⇒x = 0, 3 x = 51

⇒x = 0, x = [tex]\frac{51}{3}[/tex]

⇒x = 0, x = 17

∴ we get

When No revenue is generated, The price of the ticket is $17

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