SOLUTION: Cube A is inscribed in sphere B, which is inscribed in cube C. If the sides of cube A have length 4, what is the volume of cube C?

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Question 1155822: Cube A is inscribed in sphere B, which is inscribed in cube C. If the sides of cube A have length 4, what is the volume of cube C?
Found 2 solutions by ikleyn, jim_thompson5910:
Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
.

If cube A is inscribed in sphere B, then the diameter of the sphere B is the length 

of the longest 3D-diagonal of the cube  sqrt%284%5E2+%2B+4%5E2+%2B+4%5E2%29 = 4%2Asqrt%283%29.



Since the sphere B is inscribed in cube C, the measure of the cube C edge is  4%2Asqrt%283%29.



Hence, the volume of the cube C is  %284%2Asqrt%283%29%29%5E3 = 4%5E3%2A3%28sqrt%283%29%29 = 64%2A3%2Asqrt%283%29 = 192%2Asqrt%283%29.    ANSWER

Solved.


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

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:

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: 192*sqrt(3) cubic units
Approximate Answer: 332.553755053224 cubic units