SOLUTION: If a vendor charges p dollars each for rugby shirts, then he expects to sell 2000 -100p shirts at a tournament. a) Find a polynomial R(p) that represents the total revenue whe

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Question 203083: If a vendor charges p dollars each for
rugby shirts, then he expects to sell 2000 -100p shirts at
a tournament.
a) Find a polynomial R(p) that represents the total revenue
when the shirts are p dollars each.
b) Find R(5), R(10), and R(20).
c) Use the bar graph to determine the price that will give
the maximum total revenue.

Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
a) Revenue = (numbers of shirts)*(Price per shirt)
You are given that that price per shirt is "p" and that the number of shirts is (2000-100p). So

b)

c) R(p) is a parabola. The standard form of the equation of a parabola is: . Rewriting R(p) into this form we get: . Since the "a" (the coefficient of the squared term) is negative, the parabola opens downward. That makes the vertex of the parabola the maximum point for R(p). So we need to find the vertex of the parabola. The x-value of the vertex of a parabola is (-b/(2a)). The p-value (since yours is a function of p) of the vertex of your parabola will be .
So the maximum revenue will occur when p = 10. (And, since we already figured out R(10) in part b, we know that the maximum revenue will be $10000. But the question was what price gives the maximum revenue so the answer is 10, not 10000.)