F.3 maths problem (difficult)

a)

設錐頂是 O , 沿 OA 剪一刀 , 把 cone 的曲面展開成一扇形 OAA' ,

則 弧AA' = 底圓周 = 2π(40)

扇張角 = ㄥAOA' = [ 2π(40) / (2π(100)) ] x 360 = 2/5 x 360 = 144°

Shortest path = 直線 AA' , 而

(AA' / 2) / 100 = sin (144/2)°

AA' = 200 x sin72° = 190.2113

= 190 cm (3 sig. fig.)

b)

No.

In this case , ㄥAOA' will bigger then 144° since
the arc AA' = 2π(50) > 2π(40)
and hence AA' = the shortest path will longer then 190 cm.
(大孤對大弦)




2010-06-07 17:48:34 補充：
(大弧對大弦)

畫個扇形出黎，就會發現最頭係一條直線，即係200cm。
如果轉做50cm，結果都係一樣。