2 題 calculus

[匿名]

2011-11-27 6:50 am

1. A manufacturer has been selling 1650 television sets a week at $540 each.A market survey indicates that for each $20 rebate offered to a buyer,the number of sets sold will increase by 200 per week.
a) Find the demand function
圖片參考：http://r140.n100.queensu.ca:8081/wwtmp/equations/fe/7ca85459d2390dbf4a5dfdd0b8b8e91.png
is the number of thetelevision sets sold per week.
b) How large rebate should the company offer to a buyer, in order tomaximize its revenue?
c) If the weekly cost function is
圖片參考：http://r140.n100.queensu.ca:8081/wwtmp/equations/95/a2a7e51acbad5eabbf72a13fe0c6611.png
, how should it setthe size of the rebate to maximize its profit?

2. An ostrich farmer wants to enclose a rectangular area and then divide it intothree pens with fencing parallel to one side of the rectangle(see the figure below). He has 930 feet of fencing available tocomplete the job. What is the largest possible total area of thethree pens?

回答 (1)

用戶10012

2011-11-27 4:40 pm

✔ 最佳答案

1)
Let x = No. of TV sold per week.
Amount of rebate = $20 x (Quantity of TV sold - 1650)/200
e.g. If rebate = $20, Quantity sold = 200 + 1650 = 1850
If rebate = $40, Quantity sold = 2 x 200 + 1650 = 2050 etc.
That is amount of rebate = $20(x - 1650)/200 = (x - 1650)/10
So unit price of TV = 540 - (x - 1650)/10
Therefore, revenue p(x) = x[540 - (x - 1650)/10] = x(5400 - x + 1650)/10
= x(7050 - x)/10
dp/dx = (x/10)(-1) + (7050 - x)/10 = - x/10 + 705 - x/10 = 705 - x/5
Put it = 0, 705 - x/5 = 0, so x = 3525
That is revenue is at a max. when x = 3525, so rebate = 20(3525 - 1650)/200 = $187.50. If rebate must in steps of $20, so rebate = $180 for revenue to be max.
c)
Profit, A = Revenue - cost = x(7050 - x)/10 - 148500 + 180x
dA/dx = 705 - x/5 + 180 = 0
x = 885 x 5 = 4425 for profit to be max.
so rebate = $20(4425 - 1650)/200 = $277.50 = $280 in steps of $20.

2011-11-27 08:42:53 補充：
Correction : A should be = x(7050 - x)/10 - 148500 - 80x, so x = 5(705 - 180) = 2625 for profit to be max. Rebate = 20(2625 - 1650)/200 = $97.5 = $100.

2011-11-27 11:25:02 補充：
2) Let length of a parallel fence be x and the other side of the rectangle be 3y. So total length of fence = 6y + 4x = 930. Area of rectangle, A = 3xy = x(3y) = x(465 - 2x). dA/dx = 465 - 4x = 0, so x = 465/4, so y = (930 - 465)/6 = 465/6 = 155/2. So largest area = 3xy = 3(465/4)(155/2) = 27028

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