Two equal solid spheres are carved out from a solid cube of side length (12 + 4√3) centimetres. The spheres have the maximum possible radius and do not overlap. What is the maximum volume, in cubic centimetres, of each sphere? (Use pi = 22 / 7)

Introduction / Context:
This is a challenging three dimensional optimisation problem. Two equal spheres are to be carved out of a cube, and we want each sphere to have the largest possible radius so that both fit inside without overlapping. The side length of the cube is given in a special form (12 + 4√3) centimetres, which hints at a neat algebraic relationship. Once we find the maximum possible radius of each sphere, we calculate its volume using the standard sphere volume formula with pi = 22 / 7.

Given Data / Assumptions:

• Side length of the cube a = 12 + 4√3 cm.
• Two equal spheres are carved out from the cube.
• The spheres do not overlap and have maximum possible radius.
• Each sphere is tangent to three mutually perpendicular faces of the cube.
• The two spheres are placed near opposite corners of the cube along a space diagonal and are tangent to each other.
• pi is to be taken as 22 / 7.

Concept / Approach:
Place the cube with one corner at the origin and edges along the coordinate axes. Let one sphere of radius r be centred at (r, r, r) so that it touches the three coordinate planes. The other sphere, also radius r, is centred at (a − r, a − r, a − r) and touches the three faces at the opposite corner. For both spheres to just fit and touch each other without overlap, the distance between their centres must equal 2r. This distance lies along the space diagonal of the cube. Using the distance formula and equating to 2r gives an equation in r and a, which we solve. Then volume of each sphere is (4 / 3) * pi * r^3.

Step-by-Step Solution:
Step 1: Let the cube side length be a = 12 + 4√3 cm. Step 2: Place centres of the two equal spheres at C1 = (r, r, r) and C2 = (a − r, a − r, a − r). Step 3: Distance between the centres is |C1C2| = √[(a − 2r)^2 + (a − 2r)^2 + (a − 2r)^2] = √[3(a − 2r)^2] = √3 * (a − 2r). Step 4: For the spheres to just touch each other, this distance must equal 2r, so √3 * (a − 2r) = 2r. Step 5: Substitute a = 12 + 4√3 and solve for r: √3 * (12 + 4√3 − 2r) = 2r. Step 6: Rearrange as √3 * a − 2√3 * r = 2r, so √3 * a = 2r + 2√3 * r = 2r(1 + √3). Step 7: Thus r = (√3 * a) / [2(1 + √3)]. Factor a: a = 4(3 + √3). Step 8: Substitute a into r: r = (√3 * 4(3 + √3)) / [2(1 + √3)] = (4√3(3 + √3)) / [2(1 + √3)]. Step 9: Simplify (3 + √3) = (1 + √3) * √3, so r = (4√3 * (1 + √3) * √3) / [2(1 + √3)] = (4 * 3) / 2 = 6 cm. Step 10: Volume of each sphere is V = (4 / 3) * pi * r^3 = (4 / 3) * pi * 6^3 = (4 / 3) * pi * 216 = 288 * pi. Step 11: With pi = 22 / 7, V = 288 * 22 / 7 = 6336 / 7 ≈ 905.14 cubic centimetres.

Verification / Alternative check:
To verify, check that two spheres of radius 6 cm fit inside the cube. The full length of the space diagonal of the cube is a√3 = (12 + 4√3)√3. The centres are separated by 2r = 12 cm. The corner to corner diagonal distance must be larger than 12 plus twice the radius margin to the cube faces, which is consistent with a carefully chosen value of a. We also see that r is derived algebraically without approximation, and only the final numerical volume uses pi ≈ 22 / 7, giving about 905.14 cm^3, which matches option B.

Why Other Options Are Wrong:
The values 1077.31, 966.07 and 1007.24 correspond to incorrect assumed radii or misapplied distance relations between the centres, for example treating the spheres as if they were inscribed individually in the cube without considering mutual contact. The value 845.00 is another distractor that does not come from the exact radius r = 6. Only 905.14 matches the exact expression 288 * (22 / 7) derived from the correct geometric configuration.

Common Pitfalls:
A frequent error is to assume that each sphere is simply inscribed in the whole cube, giving radius a / 2, which would allow only one sphere, not two. Others place the spheres side by side along an edge and use a / 4 as the radius, ignoring the possibility of using the space diagonal to fit larger spheres. Another pitfall is to mis-handle the algebra when simplifying r = (√3 * a) / [2(1 + √3)]. Recognising the factorisation (3 + √3) = (1 + √3)√3 is crucial to obtaining the clean radius r = 6 cm.

Final Answer:
The maximum volume of each sphere is approximately 905.14 cubic centimetres.