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Cutting Contrives Conical Cup (Posted on 2005-04-01) Difficulty: 3 of 5
Out of a circular piece of paper, you wish to form a cone cup, so you cut out a circle wedge (with its extreme at the circle center) and join the resulting straight sides, forming a conical cup.

What size should the wedge be, to maximize the capacity of the cone?

See The Solution Submitted by Old Original Oskar!    
Rating: 4.0000 (2 votes)

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Solution Very rough approximation | Comment 1 of 17
If my math is right...

Volume of a cone = 1/3 x PI x r^2 x h

Basically we want to maximize the area of the right triangle formed by h and r, so some quick spreadsheet calculations indicate that if we maintain a hypotenuse of 1 and vary the height of the triangle from 0 to 1 in .01 increments, we get a maximum area where h=0.58 and r=0.8146.

The circumference of the base of the cone is 2 x PI x r = 5.1183819, and the circumference of the original circular cutout is 6.28318.  Dividing the 5.1183819 value by the 6.28318 value wee see that the area of the remaining area of the circle (after we cut the wedge) should be 81.461647% of the area of the original circle.  Multiplying this 81.5% by 360 in a circle we end up with an angle for the cone section of 293.26.

Subtracting this angle from 360 we see that the optimum angle for the wegde cutout is 66.738.

...I think...
  Posted by Erik O. on 2005-04-01 18:37:22
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