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?

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

reduce to get V=1/3ð*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-è/360)  where è= the angle of the wedge

è=360(1-SQRT(2/3)) = 66.06 deg 

 

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