Let
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
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.