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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)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Very rough approximation | Comment 3 of 17 |
(In reply to Very rough approximation by Erik O.)

I did it a little differently.  Since the radius of the circle doesn't change the answer and to make things simple I let the radius =1.  and I want an expression without h in it so: using a right angle triangle I got:  R=r^2+h^2    ;where R=radius of circle=1 ==> h=SQRT(1-r^2)    substitute in

V =1/3r^2h    and get    V=1/3r^2*SQRT(1-r^2)

reduce to get V=1/3*SQRT(r^4-r^6)

Now we want to find when the rate of r changing = 0  in other words dV/dr =0    Since my calculus isn't that good I solved it for V^2 which gives the same point.

dV^2/dr=0=4r^3-6r^5

solving for r we get r=SQRT(2/3)=0.8165

r=(1-/360)  where = the angle of the wedge

=360(1-SQRT(2/3)) = 66.06 deg 

 


  Posted by Mark on 2005-04-01 19:03:47
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