A college bookstore finds that if it charges dollars for a

A college bookstore finds that if it charges ? dollars for a T-shirt, it sells (1000 20p) T-shirts. Its revenue is the product of the price and the number of T-shirts it sells. Express the revenue R(p) as a quadratic function of the price p For what prices is the revenue equal to zero? What price should it charge in order to maximize the revenue? What would the maximum revenue be? What would the revenue be if they charge $19.95 per T-shirt?

Solution

(a) The revenue R(p) at price p is given by

Revenue = R(p) = (Price)(Number sold) = p(1000 20p).

      Factoring 20 from each term in 1000 20p, we have

R(p) = 20p(50 + p) = 20p(p 50).

(b) The revenue has factors p and p50. From the first factor, we see the revenue is zero when p = 0. This makes sense since if the bookstore does not charge any money for the T-shirt, it does not earn any revenue. From the second factor, we see the revenue is also zero when p = 50. If the price of a T-shirt is p = 50, then the number of T-shirts the bookstore sells is 100020(50) = 0. The T-shirts are so expensive that nobody is willing to buy them.