amaths --differentiation ....

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2008-03-28 6:51 am

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A piece of wire of length 20 cm is cut into two pieces, one of which is bent to form a circle and the other to form a square. Let x cm be the length of the square and A cm2 be the total area enclosed by the square and the circle.
(a) Express A in terms of x and find the range of values of x.
(b) Find the range of values of x for which
(i) dA/dx < 0
(ii) dA/dx > 0

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回答 (1)

慶哥

2008-03-28 10:09 am

✔ 最佳答案

(1)Lenght of wire of the square=4x
Lenght of the wire of the circle=20-4x
Since 20-4x>0,0＜x＜5

Let r be the radiu of the circle,2派r=20-4x
r=(10-2x)/派
A=x^2＋派r^2
A=x^2＋[(10-2x)^2]/派
A=x^2＋[4x^2-40x＋100]/派
A=x^2＋(4/派)x^2-(40/派)x＋100/派

dA/dx=2x＋(8/派)x-40/派

(i)dA/dx＜0,
2x＋(8/派)x-40/派＜0
0＜x＜20/(派＋4)

(ii)dA/dx＞0
20/(派＋4)＜5
派＝圓周率

2008-03-28 02:10:22 補充：
(1)Lenght of wire of the square=4x
Lenght of the wire of the circle=20-4x
Since 20-4x＞0,0＜x＜5

2008-03-28 02:11:12 補充：
(ii)dA/dx＞0
20/(派＋4)＜x＜5
派＝圓周率

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