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.
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
You have made a good start in setting up the equation. Let's continue solving it step by step.
-2x^2 + 2400x = 0
To find the maximum income, we need to find the vertex of the quadratic equation. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation (in the form ax^2 + bx + c = 0).
In our equation, a = -2 and b = 2400, so substituting these values into the formula, we get:
x = -(2400) / (2(-2))
x = -2400 / -4
x = 600
So, the x-coordinate of the vertex is 600. Now, to find the maximum income, we need to substitute this value of x back into the equation I(x) = -2x^2 + 2400x:
I(600) = -2(600)^2 + 2400(600)
I(600) = -2(360000) + 1440000
I(600) = -720000 + 1440000
I(600) = 720000
Therefore, the maximum income is $720,000.
Now, let's move on to the second part of the question and find the ticket prices for which there will be no income. We can determine this by setting the income function equal to zero:
-2x^2 + 2400x = 0
Factor out x from both terms:
x(-2x + 2400) = 0
This equation will be true if either x = 0 or -2x + 2400 = 0.
Solving -2x + 2400 = 0:
-2x = -2400
x = -2400 / -2
x = 1200
Therefore, there will be no income when the ticket price is either $0 or $1200.