Question 199322
A  rough  sketch  will  greatly  aid  in  solving  this
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diagonal  of  the  cube  is  dia ^2 = 3 * 10 ^2,,,,diagonal  is  17.320
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the  largest  sphere in  the  box  has  a  dia  of  10,,,sides  of  the  box
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alone  the  diagonal,  the  distance  left  between  sphere (r=5) and  corner(diag = 17.32)
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is  (17.32 - 10 ) /2 = 3.6602.
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now  comes  the  hard  part,  a  sphere  of  that  dia  will  not  fit,  it  goes  to  corner.
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to  calculate  a  sphere that  touches  the  sides  of  the cube, remember  that  3.6602  is
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also  the  diagonal  of  a  small  box, that  would  fit  in  corner,  with  a  small  sphere  inside.
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if  the  diagonal  of  the  box  is  3.6602,  but  s^2 +s^2 +s^2 = diag^2,,,s= sqrt ( diag^2 /3)
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s = sqrt 3.6602^2 /3 = 2.11,,,the  diameter  of  the  sphere