Question 1155822
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Edit: I realized I made an error, but I have fixed the solution below (this is more or less a complete rewrite compared to my original solution).


Diagram of cube A:
<img src = "https://i.imgur.com/dp1xJ2u.png">
The left shows a 2D flat view of the bottom face of the cube. This is square ABCD. The right shows the 3D version of cube A. At the center of cube A is point K


I've added point M directly below point K such that M is on the bottom face of the cube. Point L bisects segment AB. Triangle ALM has legs of 2 each, so the hypotenuse is 2*sqrt(2) through the pythagorean theorem. So AM = 2*sqrt(2)


Triangle AMK has legs AM = 2*sqrt(2) and MK = 2. Using the pythagorean theorem again has us get AK = 2*sqrt(3). Therefore, segment AH = 2*(2*sqrt(3)) = 4*sqrt(3). You can use the space diagonal formula or the distance formula to find the length of AH. 


The radius of sphere B is 4*sqrt(3) units long.
The side length of cube C is 4*sqrt(3) units long. This is so sphere B fits snugly inside cube C. 


Volume of cube = (side length)^3
Volume of cube C = (4*sqrt(3))^3
Volume of cube C = (4*sqrt(3))(4*sqrt(3))(4*sqrt(3))
Volume of cube C = [(4*sqrt(3))(4*sqrt(3))](4*sqrt(3))
Volume of cube C = (4*4*sqrt(3)*sqrt(3))(4*sqrt(3))
Volume of cube C = (16*3)(4*sqrt(3))
Volume of cube C = 48(4*sqrt(3))
Volume of cube C = 192*sqrt(3)
Volume of cube C = 332.553755053224
Use your calculator to compute the approximate value in the last step.




Exact Answer: <font color=red>192*sqrt(3) cubic units</font>
Approximate Answer: <font color=red>332.553755053224 cubic units</font>
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