F.3 maths problem (difficult)

a)
設錐頂是 OA , 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 = 2π(50) > 2π(40)
and hence AA' = the shortest path will longer then 190 cm.