Question

At a football game, the number of tickets sold is affectted by the ticket price. They expect the income to be l(x) dollars as a function of of the ticket price in dollars will be given by I (x) = –2x ^ 2 + 2400x = 0

1) Calculate their max income.
2) For which ticket prices will there be no income at all?


I'm pretty sure I have to do it like to start:
-2x^2+2400x=0
(–2x^2)/(-2) + (2400x)/(-2) = 0/(-2)
x^2 - 1200x = 0
x*(x - 1200) = 0

But I'm not sure if that's right or how to continue calculations.

Answer 1

You have I(x) = -2x^2 + 2400x
This is a parabola which opens downwards, so the vertex would give you
the maximum I
Don't know which method you have learned to find the vertex of a parabola
the quickest way would be to use ...
the x of the vertex is -b/(2a) = -2400/-4 = 600
So the price per ticket yielding the greatest income is $600 per ticket
and that income is
I(600) = -2(600^2) + 600(2400) = $720,000

by setting -2x^2 + 2400x = 0 and showing what you did above,
you are solving part b)

-2x^2 + 2400x = 0
-2x(x - 1200) = 0
so x = 0 , clearly when a ticket costs nothing, there won't be any income
or
x = 1200
I guess at $1200 per ticket they are not going to sell any, makes sense

The following shows my answers are correct
https://www.wolframalpha.com/input/?i=graph+y+%3D+-2x%5E2+%2B+2400x