Problem: Rosalie is organizing a circus performance to raise money for a charity. She is trying to decide how much to charge for tickets. From past experience, she knows that the number of people who will attend is a linear function of the price per ticket. If she charges $5, 1200 people will attend. If she charges $7, 970 people will attend. How much should she charge per ticket to make the most money?
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If you consider these numbers as points, where x = ticket price and y = people then you have
P1 = (4, 1200) and P2 = (7, 970)
Determine the equation of a line that passes through these points. You will obtain a linear equation of the form y = mx + b.
Notice, though, that Profit itself is a function of x and y. In fact Profit = x * y (it equals the number of people multiplied by the ticket price).
So your profit equation would be P(x) = x * y = x * (mx + b)
Determine the vertex of this profit equation. This will be the point, x (the ticket price) that results in the maximum profit P(x).
What is the exact equation?
I meant to say P1 = (5, 1200) ********
Well if the points are (5, 1200) and (7, 970), the slope is (1200 - 970) / (5 - 7) = -115
In point slope form we can write y - y1 = m(x - x1) y - 970 = -115(x - 7)
y = -115x + 1775
Our profit is x * y or x * (-115x + 1775) Profit = P(x) = -115x^2 + 1775x
This is a parabola with a maximum value of P(x) where x is the x-coordinate of the vertex. Can you determine the vertex of this parabola?