F.4 Maths quadratic equation

yuet

2009-11-14 5:09 am

A sausage-shop sells German sausages costing $2 each, at a selling price of $5, and the quantity of German sausages sold per day is 200.The shop wants to promote the sales by reducing the price. It is known that for every decrease of $0.1 in the selling price of German sausage, the quantity of sausages sold will increaseby 10.
(a) Suppose the selling price of German sausage is decreased by $x, and the total profit made per day ins selling German sausages is $P. Express P in terms of x.
(b) what should the selling price of German sausage be in order to get the maximum total profit? What is the maximum total profit per day in selling german suasages?

回答 (1)

用戶10012

2009-11-14 5:34 am

✔ 最佳答案

(a) New selling price = (5 - x)
Selling quantity increase = (x/0.1)(10) = 100x
So total quantity = 200 + 100x
Profit, P = [(5 - x) - 2](200 + 100x) = (3 - x)(200 + 100x)
= 100(3 - x)(2 + x)
(b)
P = 100(3 -x)(2 + x)
= 100(6 + x - x^2)
= -100(x^2 - x - 6)
= -100[(x - 1/2)^2 - 1/4 - 6]
= -100(x - 1/2)^2 + 25 + 600
= -100(x - 1/2)^2 + 625
so profit is max. when selling price = 5 - 1/2 = 4.50
Max. profit = $625.

2009-11-13 21:38:30 補充：
Profit is max. when x = 1/2, that means when selling price = 5 - x = 5 - 1/2 = 4.50

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