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.
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