F4 Maths 3-D problems -.-

Anonymous

2010-03-21 12:17 am

In the figure, the length of each leg of a tripod for cameras is 130 cm.
Angle HAD between each leg and the height ( AH ) of the tripod above
the horizontal ground can be adjusted from 25 degree to 40 degree.
Find

The range of the angle between AB and AD.

Figure: http://img221.imageshack.us/img221/6007/maths27.jpg

回答 (1)

用戶2053

2010-03-21 1:20 am

✔ 最佳答案

圖片參考：http://img221.imageshack.us/img221/6007/maths27.jpg

ㄥDAH = θ , (25° <= θ <= 40°)

ㄥHDB = 60/2 = 30°

;;

HD / 130 = sin θ

i.e. HD = 130 sin θ

(BD/2) / HD = cos30°

BD / 2HD = √3 / 2

BD = (√3)HD

BD = 130√3 sinθ

cosㄥBAD = (AB^2 + AD^2 - BD^2) / (2 AB AD)

cosㄥBAD = (130^2 + 130^2 - (130√3 sinθ)^2) / (2 * 130 * 130)

cosㄥBAD = [33800 - 50700 (sin θ)^2] / 33800

Since sin 25° <= sin θ <= sin 40°

The max. of cosㄥBAD = [33800 - 50700(sin 25°)^2] / 33800

= 0.73209 , min. ㄥBAD = 42.938°

The min. of cosㄥBAD = [33800 - 50700(sin 40°)^2] / 33800

= 0.38024 , max. ㄥBAD = 67.652°

42.938° <= θ <= 67.652°

2010-03-20 17:23:13 補充：
42.938° <= θ <= 67.652°

should be

42.938° <= ㄥBAD <= 67.652°

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