Question

Profit Parabolas: The baseball card shop is trying to sell limited baseball cards for $30, yet no one is interested in buying the cards. The owner of the baseball card shop, reduced the price of the cards to help sales. The baseball card owner found for every $1 that, the price was reduced, it created 8 more customers daily( per day). If the baseball card owner reduced the price to $ 20, how many people(customers) will be buying baseball cards each day? What will be the total profit? What should be the price of the baseball cards to create the greatest profit for the owner? Graph the function. Clearly label the graph.

Answer 1

There is an issue with the question. Since the cost of baseball cards is not given, the total daily profit cannot be determined. Therefore, we presume that it is the total revenue which is required.

The baseball cards being sold at $ 30 do not elicit any response. However, for every reduction in price by $ 1, 8 more customers are buy baseball cards per day. If the baseball card owner reduced the price to $ 20 i.e. by $ 10, the number of customers who will be buying baseball cards each day is 8*10 = 80. The total revenue will be 80*$ 20 = $ 1600. However, since the cost of baseball cards is not given, the total daily profit cannot be determined.

Let the reduction in the price of baseball cards by $ x per card create the greatest revenue for the owner. Then the revenue is R(x) = price* number of cards sold = (30-x)(8x) = -8x² +240x = -8(x² -30x ) = -8[x² -2x*15 +(15)²]+8*(15)² = -8(x-15)² +1800. This is the equation of a downwards opening parabola with vertex at (15,1800). Thus, the greatest daily revenue for the owner is $ 1800. The price of the baseball cards to create the greatest profit for the owner is $ (30 -15) = $ 15.

A graph of R(x) = -8x² +240x is attached.

Note: The coordinates of the vertex, i.e. (15,1800) can be seen only in the original desmos graph and not in an image.