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Even more coni-mising

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Even more impressed, the High King Math decides to grant you one final reward. He hands you a circle with radius R of some flexible, heat resistant material, which you may cut a slice out of and then roll into a (right circular) cone, with which he will mold you a final cone of gold. How much do you cut?

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Cut a slice having angle PHI so that a fraction f = (2pi-PHI)/2pi of the disc remains.

Make the cone.

Its radius r = f R

Its height h = sqrt (R2 - r2) = R sqrt (1 - f2)

Its volume v = (pi/3) h r2 = (pi/3) R3 f2 sqrt (1 - f2)

f2 sqrt (1 - f2) has a maximum of 2 sqrt (3) / 9 when f = sqrt (2/3) and PHI = 1.153 rad = 66.06o

vmax = 2 pi R3 sqrt (3) / 27 = ~0.403 R3 .

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