F.5 maths inequalities

2011-04-16 4:07 am

In the figure, ABCD is a cyclic quadrilateral where AB = AD and BC = DC. It is given that the perimeter of ABCD is 20cm. If the area of the shaded region
is not greater than 50 cm2, find the maximum length of BC.
圖片參考：http://imgcld.yimg.com/8/n/HA00259508/o/701104150095913873426850.jpg

更新1:

find the maximum length of BC*

回答 (1)

myisland8132

2011-04-16 6:50 am

✔ 最佳答案

△ABC ≅ △ADC =>∠B = ∠D = 90 =>AC is diameter.

AB^2 + BC^2 = AC^2. But AB + BC = 10 => AB = 10 - BC

So, (10 - BC)^2 + BC^2 = AC^2 or 2BC^2 - 20BC + 100 = AC^2

Since area of shaded region

= πAC^2/4 - (1/2)(AB)(BC)

= πAC^2/4 - (1/2)(10 - BC)(BC)

= π[(1/2)BC^2 - 5BC +25)] - (1/2)(10 - BC)(BC)

= (1/2)(π - 1)BC^2 - 5(π + 1)BC + 25π

Consider (1/2)(π - 1)BC^2 - 5(π + 1)BC + 25π <= 50

(π - 1)BC^2 - 10(π + 1)BC + 50(π - 2) <= 0

1.4936 <= BC <= 17.8453

So, the maximum length of BC is 17.8453 cm

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