Ghost of Guldinus
A solid circumsphere encloses a Reuleaux triangle spheroform hollow void that contains it's own solid insphere. If a solid insphere that can fill the hollow void has an unit inradius, find the proportion of the solid mass total the solid insphere can constitute.
Provide your answer only as an integer percentage.
Note: The radii of the largest inscribed circle of a Reuleaux triangle of width " s ", and of the circumscribed circle of the same triangle, are 0.42265 s & 0.57735 s respectively.
The volume formula of a Reuleaux triangle spheroform of width " s " = 0.44946 s^3 .
The answer is 47.
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1 solution
Inradius of insphere = r = 1
Volume of insphere = v = ( 4 / 3 ) x pi x r^3 = ( 4 / 3 ) x pi x 1^3 = 4.189
Radius of void = s = r / ( 0.42265 x { s } ) = 1 / ( 0.42265 x { s } ) ; s = 1 / 0.42265 ; s = 2.366
Volume of void = S = 0.44946 x { s^3 } = 0.44946 x 2.366^3 = 5.953
Circumradius of circumsphere = R = 0.57735 x { s } = 0.57735 x 2.366 = 1.366
Volume of circumsphere = V = ( 4 / 3 ) x pi x R^3 = ( 4 / 3 ) x pi x 1.366^3 = 10.676
Percentage of insphere mass = { v / [ ( V - S ) + v ] } x 100
Percentage of insphere mass = { 4.189 / [ ( 10.676 – 5.953 ) + 4.189 ] } x 100 = 4.189 / ( 4.723 + 4.189 ) x 100 = ( 4.189 / 8.912 ) x 100 = 47%