A bookstore can obtain a certain gift book from the publisher at a cost of $3 pe

Question

A bookstore can obtain a certain gift book from the publisher at a cost of $3 per book. The bookstore has been offering the book at the price of $18 per copy, and at this price, has been selling 200 copies a month. The bookstore is planning to lower its price to stimulate sales and estimates that for each $1 reduction in the price, 20 more books will be sold each month. Express the bookstore's monthly profit from the sale of this book as a function of the selling price, draw the graph and estimate the optimal selling price.

A bookstore can obtain a certain gift book from the publisher at a cost of $3 per book. The bookstore has been offering the book at the price of $18 per copy, and at this price, has been selling 200 copies a month. The bookstore is planning to lower its price to stimulate sales and estimates that for each $1 reduction in the price, 20 more books will be sold each month. Express the bookstore's monthly profit from the sale of this book as a function of the selling price, draw the graph and estimate the optimal selling price.

Explanation / Answer

Sales = 200 + 20x this says for each price decrease (x) my sales will increase by 20
Profit/Sale = 17 - x this says my profit is Price(20) - Cost(3) - 1 for each price decrease
Profit = Sales * Profit/Sale = (200 + 20x) * (17-x)
3400 +140x - 20x^2

take first derivative = 140 - 40x and set = 0, take second derivative (-40), if < 0 we have a local max
solve for x when first derivative = 0
140(140/40-x) = 0
x = 3.5

sale price = $16.5

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