For the demand equation, express the total revenue r as a function of the price p per item.
q = −5p + 500
r(p) = ?

Revenue, R = quantity sold * price

price = p

quantity sold = q = -5p + 500

R(p) = q*p = (-5p + 500) * p = -5p^2 + 500p

R(p) = -5p^2 + 500p

That is a quadratic function (parabola), which opens downward, so it has a maximum value, which represents the maximum revenue that can be obtained.

That point is the vertex of the parabola. It tells the price and the number of items sold to obtain the greatest revenue. You can obtain the vertex by several ways.

One is finding the roots of the quadratic function; then the vertex is at the middle of the roots.

-5p^2 + 500p = -5p(p-100) => p = 0 and p =100 => vertex is at p = 50 and maximum R = -5(50^2) + 500(50) = 12,500

 

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