 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Cutting Contrives Conical Cup (Posted on 2005-04-01) 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) Comments: ( Back to comment list | You must be logged in to post comments.) Straight forward (if dull) approach | Comment 11 of 17 | Isn��t 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 simplicity��s 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 �� = 2�С�(2/3) (maximum)

Thus the angle of the wedge to be removed

= 2��-��

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

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