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 Math Central Quandaries & Queries
 Question from Shawn, a student: Your neighbours operate a successful bake shop. One of their specialties is a cream covered cake. They buy them from a supplier for $6 a cake. Their store sells 200 a week for$10 each. They can raise the price, but for every 50cent increase, 7 less cakes are sold. The supplier is unhappy with the sales, so if less than 165 cakes are sold, the cost of the cakes increases to $7.50. What is the optimal retail price per cake, and what is the bakeshop's total weekly profit? Shawn, I assume that, if you are facing a question like this, you have basic calculus skills or are prepared to use graphical solution methods. First, let x be (P-10) where P is the price per cake in dollars. The number of cakes sold is 200-14x. What is the profit per cake (a) at$6 per cake and (b) at \$7.50 per cake wholesale, as a function of x?

What is the total profit in each case? Graph each of these lightly, then shade each curve heavily over the region where it describes the actual situation (are sales above or below 165 cakes?) The heavily-shaded curve is the profit function.

Now locate the local maxima of the profit function, which may occur either where a derivative is zero or at the end of a section of the graph. The largest of these is the global maximum, which is what you want.

Good Hunting!
RD

PS: It is not clear to me that the business described should be described either as a "bake shop" [if they don't bake their own specialties] or "successful" [if their supplier is considering penalizing them for poor sales.]

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