紙碟與體積

A paper disc can be ' folded ' to become a cone by ' folding ' it into a circular sector first.
Let x be the angle of the sector in radian.
Radius of sector = a = 17.78/2 = 8.89 cm.
Arc length = ax = circumference of the base circle of cone = 2πr
so r = ax/2π.
Let height of cone = h, by Pythagoras thm.,
a^2 = h^2 + (ax/2π)^2
so h = sqrt [ a^2 - (ax/2π)^2 ] = a sqrt [ 1 - (x/2π)^2 ]
Volume of cone, V = aπ(ax/2π)^2 sqrt [ 1 - (x/2π)^2 ]/3
= a^3x^2/12π sqrt [ 1 - x^2/4π^2 ]................. (1)
dV/dx = a^3/12π { 2xsqrt [ 1 - x^2/4π^2 ] + x^2(-2x/4π^2)/2 sqrt [ 1 - x^2/4π^2 ]}
Put dV/dx = 0
4x ( 1 - x^2/4π^2) - x^3/2π^2 = 0
8π^2x( 1 - x^2/4π^2) - x^3 = 0
8π^2x - 2x^3 - x^3 = 0
x ( 8π^2 - 3x^2) = 0
x = 0, +/- π sqrt(8/3)
So volume of cone is maxmum when the angle of the ' folded ' sector is
π sqrt (8/3) radians or 294 degree.
Sub. this value of x into (1), the maximum volume is 283.2 cm^3.

你應該是指：

1. 圓紙碟（直徑 7 吋）
2. 問最大容量、非體積

這不單單是一個數學題，因為如果原紙版是圓形，摺疊時會有重覆的地方。
你要提及如對待握疊的地方才行。

又例如你有否要求所摺出的立體需要是圓底等等。

2014-07-30 09:38:52 補充：
立體是否需要圓底？

如果是的話，那相信本題能夠順利解答。

(◕‿◕✿)

2014-07-30 11:33:56 補充：
哦？

你要摺成空心圓錐而不是空心圓柱？