SOLUTION: A baseball team plays in a stadium that holds 55,000 spectators. With the ticket price at $10, the average attendance at recent games has been 27,000. A mar- ket survey indicat

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: A baseball team plays in a stadium that holds 55,000 spectators. With the ticket price at $10, the average attendance at recent games has been 27,000. A mar- ket survey indicat      Log On


   

Question 1189985: A baseball team plays in a stadium that
holds 55,000 spectators. With the ticket price at $10, the
average attendance at recent games has been 27,000. A mar-
ket 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.
Found 2 solutions by Solver92311, ikleyn:
Answer by Solver92311(821) About Me  (Show Source):
You can put this solution on YOUR website!


Let represent the number of dollars of reduction in the ticket price. So the ticket price is represented by and the average attendance is represented by

Revenue is ticket price times attendance. You should be able to do the rest.

John

My calculator said it, I believe it, that settles it

From
I > Ø

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
A baseball team plays in a stadium that
holds 55,000 spectators. With the ticket price at $10, the
average attendance at recent games has been 27,000. A mar-
ket 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.
~~~~~~~~~~~~~~~~ 

The problem tells us that if the ticket price is $10, then the attendance is 27000

and that ticket price change of $1 produces the attendance change of 3000 in opposite direction

    (when the price goes down, the attendance goes up, 
     and vice versa: when the price goes up, the attendance goes down).     (*)


In addition, it says that the attendance is a linear function of price.


It means that the slope of the plot is -3000, so we can write the attendance 
as a linear function of the ticket price in this form

        A(p) = 27000 - 3000*(p-10).


Indeed, this function is linear and satisfies the pointed properties (*).


Now, the revenue is the product of the ticket price by the attendance

    R(p) = p*A(p) = p*(27000 - 3000*(p-10)) = p*(27000 - 3000p + 30000) = p*(57000 - 3000p) = -3000p*2 + 57000p.


ANSWER.  Under given conditions, the revenue function is this quadratic function of the ticket price

                R(p) = -3000p*2 + 57000p.


CHECK.   At the ticket price of $10,  the revenue is  R(10) = -3000*10^2 + 57000*10 = -300000 + 570000 = 270000 dollars.

         Compare it with 27000*10 = 270000: these numbers coincide.


The given formula is valid until the attendance is not greater than the maximum capacity of the stadium of 55000 spectators.

Solved.