The volume of the largest cone that can be inscribed inside a unit sphere is of the form for coprime positive integers and . What is ?
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Using the formula formula for circumradius of an isosceles triangle of base a and height h :
R = 8 1 ( h a 2 + h )
Taking a section of the sphere along its central axis, we will have a triangle whose base is the cone base diameter and whose height is the cone height, which have to be circumscribed by a circle of radius 1. Thus, denoting cone radius as r and height as h :
1 = 8 1 ( h 4 r 2 + h )
r 2 = 2 h − h 2
So cone volume is:
v = 3 π r 2 h = 3 π ( 2 h 2 − h 3 )
Differetiating v with respect to h and making it equal to 0 :
3 π ( 4 h − 3 h 2 ) = 0
h = 3 4
Plugging this in v :
v = 3 π ( 2 ⋅ 9 1 6 − 2 7 6 4 )
v = π ⋅ 8 1 3 2
Hence, A = 3 2 , B = 8 1 and A + B = 1 1 3