Question 953226
The volume of a sphere is:
{{{ V[s] = (4/3)*pi*r^3 }}}
where {{{ r }}} is the radius
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You are given that:
{{{ V[s] = 413 }}} cm3
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{{{ 413 = (4/3)*r^3 }}}
Multiply both sides by {{{ 3 }}}
{{{ 1239 = 4*pi*r^3 }}}
Divide both sides by {{{ 4*pi }}}
{{{ 309.75 / pi = r^3 }}}
{{{ r^3 = 98.5966 }}}
{{{ r = 4.6198 }}}
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check:
{{{ V[s] = (4/3)*pi*r^3 }}}
{{{ V[s] = (4/3)*pi*4.6198^3 }}}
{{{ V(s) = 1.333*3.14159*98.5983 }}}
{{{ V(s) = 412.997 }}}
This looks close enough
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If this sphere is tightly enclosed in a cube,
the volume of the box is {{{ V[c] =  (2r)^3 }}}
This is because {{{ 2r }}} is equal to a side
of the box
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{{{ V[c] = ( 2*4.6198 )^3 }}}
{{{ V[c] = 9.2396^3 }}}
{{{ V[c] = 788.7866 }}}
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{{{ V[c] - V[s] = 788.7866 - 413 }}}
{{{ V[c] - V[s] = 375.7866 }}} ( wasted volume in cm3 )
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You are asked for:
{{{ ( V[c] - V[s] ) / V[c] )*100 }}} which is a %
{{{ ( 375.7866 / 788.7866 )*100 }}}
{{{ .4764*100 = 47.64 }}}
47.64% is wasted space inside the cube
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Hope I got it -tough calculations