Question 203083
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
{{{R(p) = (2000-100p)p = 2000p - 100p^2}}}<br>
b) {{{R(5) = 2000(5) - 100(5)^2 = 2000(5) - 100(25) = 10000 - 2500 = 7500}}}
{{{R(10) = 2000(10) - 100(10)^2 = 2000(10) - 100(100) = 20000 - 10000 = 10000}}}
{{{R(20) = 2000(20) - 100(20)^2 = 2000(20) - 100(400) = 40000 - 40000 = 0}}}<br>
c) R(p) is a parabola. The standard form of the equation of a parabola is: {{{ax^2 + bx + c}}}. Rewriting R(p) into this form we get: {{{R(p) = -100x^2 + 2000p}}}. 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 {{{-b/(2a) = -2000/(2(-100)) = -2000/-200 = 10}}}.
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.)