A sphere has a diameter 50√3 cm. The surface area of the largest possible cube…

Concept: When a cube is inscribed in a sphere so that every vertex of the cube touches the sphere's surface, the sphere's diameter equals the cube's space diagonal. For a cube of side length a, the space diagonal is a√3, so diameter = a√3.

1. The sphere's diameter is given as 50√3 cm.

2. Equate the space diagonal to the diameter: a√3 = 50√3.

3. Divide both sides by √3 to get the side length of the cube: a = 50 cm.

4. Apply the cube’s surface area formula, 6a2: 6 × 502 = 6 × 2500 = 15000 cm2.

Cross-check: the space diagonal also satisfies (space diagonal)2 = a2 + a2 + a2 = 3a2. With a = 50, this gives 3 × 2500 = 7500, and (50√3)2 = 2500 × 3 = 7500 — the two match, confirming the side length.

So the largest cube that fits inside the sphere has a total surface area of 15000 cm2.