SOLUTION: Two spheres of equal radius are taken out by cutting from a solid cube of side 13cm. What is the maximum volume(in cm^3) of each sphere.

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Question 1114815: Two spheres of equal radius are taken out by cutting from a solid cube of side 13cm. What is the maximum volume(in cm^3) of each sphere.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

If sphere is in a cube, the of the sphere is to the length of cube’s .
Since you have 2 spheres taken out by cutting from a solid cube (they are actually inscribed in cube) then the side of the cube equal to the of their which are equal in length.
so, you have


the length of the each radius is:





the maximum volume of each sphere is:




≈

Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.
I read and interpret the condition by different way: two spherical solids are taken out by cutting from a solid cube
in a way that their centers are located on the  3D  (=longest) diagonal of the cube.



        (This condition provides the maximum radius and maximum volume to each of the two spheres).



Then it is clear that these spheres touch each other at the middle of the  3D  diagonal of the cube.


The length of the longest 3D diagonal of this cube is    cm.


If "r" is the radius of the sphere, then     =  cm.


Hence,    r =  = 4.12 cm.


Then the volume of each sphere    V =  =  = 292.8 cm^3.

Solved.



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