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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 Straight forward (if dull) approach | Comment 11 of 17 |

Isnt this easiest to solve by making the volume of the cone a function of the angle and grinding through the differentiation?

<o:p> </o:p>

Let the radius of the wedge = slope length of the cone = 1 for simplicitys sake.

Calling the angle of the paper used (in radians), , we get that the circumference of the cone = arc length of the wedge used = .  Thus the radius of the base of the cone is /2.

Pythagoras then tells us the height of the cone is (1-^2/4^2).

<o:p> </o:p>

Volume of cone, V= 1/3**r^2*h

                             = 1/3**^2/4^2* (1-^2/4^2).

                             = 1/12(^2 (1-^2/4^2).

<o:p> </o:p>

Max volume will be when dV/d = 0

<o:p> </o:p>

dV/d = 1/12[2* (1-^2/4^2) C ^3/(4^2* (1-^2/4^2))]

<o:p> </o:p>

Setting this equal to zero gives 3^3-8^2* = 0

Solving finds = 0 (minimum volume) or = 2С(2/3) (maximum)

Thus the angle of the wedge to be removed

= 2-

= 2(1-(2/3)) rad

  Posted by Alec on 2005-04-02 10:12:17
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