The radii of the bases of two right circular solid cones of the same height are r1 & r2. The cones are melted & recasted into a solid sphere if radius R . show that the height of each cone is given by h=4R^3÷r1^2+r2^2 ?

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Answer 1 · 2018-02-27T18:52:36

See below. Quite simple really.

Volume of cone 1; $pi*r_1^2*h/3$
Volume of cone 2: $pi*r_2^2*h/3$
Volume of the sphere: $4/3*pi*r^3$

So you have:

$"Vol of sphere" = "Vol of cone 1" + "Vol of cone 2"$

$4/3*pi*R^3 = ( pi*r_1^2*h/3 ) + (pi*r_2^2*h/3)$

Simplify:

$4*pi*R^3 = (pi*r_1^2*h ) + (pi*r_2^2*h)$

$4*R^3 = ( r_1^2*h ) + (r_2^2*h)$

$h = (4R^3)/ ( r_1^2 + r_2^2)$