Originally posted by iamatiger
Say the paper circle has circumference C
It is cut into two pieces, the smaller piece has circumference xC, and the larger has circumference (1-x)C
side length of both cones = Radius of paper circle = C/(2*pi())
Radius of smaller cone = xC/(2*pi)
Radius of large cone = (1-x)C/(2pi)
cone_height = sqrt(Side^2 - Radius^2)
So height of smaller c ...[text shortened]... quare it all and maximise the ratio of the squares of the volume, then differentiate a quotient?
Hmm, that doesn't seem to work, has no maxima. Ah, I see, we need the difference in the volumes, not the ratio.
volume of small cone = 1/3 * pi * ( xC/(2*pi) )^2 * sqrt((1-x^2)C^2/(4*pi^2))
volume of large cone = 1/3 * pi * ( (1-x)C/(2pi) ) ^ 2 * sqrt((2x - x^2)C^2/(4*pi^2))
large - small = K ((1-x)^2*sqrt((2x-x^2) - x^2*sqrt(1-x^2))
so we need to find the x value, between 0 and 1, that gives the maximum value of
(1-x)^2*sqrt(2x-x^2) - x^2*sqrt(1-x^2)
Pasting that into wolfram:
We get x = 0.147977, which agrees with you to, now we just need to differentiate it...
differentiate (1-x)^2*sqrt(2x-x^2) - x^2*sqrt(1-x^2)
into wolfram, and clicking on show steps seems to imply this is not a sum I really want to do by hand that much.