A solid cube has each edge of length 18 cm. A sphere is carved out so that it just fits inside the cube, touching all six faces (the largest possible inscribed sphere). Find the volume of this sphere, in cubic centimetres, leaving your answer in terms of pi.

Introduction / Context:
This three dimensional geometry problem involves a sphere inscribed inside a cube. The largest possible sphere inside a cube will touch all six faces of the cube. Therefore, its diameter equals the edge length of the cube. Once the radius is determined from the cube edge, the volume of the sphere can be found using the standard formula. The final answer is expressed in terms of pi, which is common in volume questions.

Given Data / Assumptions:

• Edge of cube = 18 cm.
• Largest inscribed sphere touches all faces of the cube.
• Diameter of sphere = edge of cube.
• Volume of a sphere V = (4/3) * pi * r^3.

Concept / Approach:
Because the sphere is inscribed, its diameter equals 18 cm. So radius r = 18 / 2 = 9 cm. Substitute r into the volume formula V = (4/3) * pi * r^3. It is convenient to compute r^3 first and then multiply by 4/3. The expression will simplify naturally, and we keep pi as a symbol to match the answer options.

Step-by-Step Solution:
Edge of cube = 18 cm, so diameter of sphere = 18 cm.Radius r = 18 / 2 = 9 cm.Volume of sphere V = (4/3) * pi * r^3.Compute r^3: 9^3 = 9 * 9 * 9 = 729.So V = (4/3) * pi * 729.Multiply 729 by 4: 729 * 4 = 2916.Now divide by 3: 2916 / 3 = 972.Thus V = 972π cubic centimetres.

Verification / Alternative check:
As a quick check, note that the volume of the cube is 18^3 = 5832 cubic centimetres. The volume of the inscribed sphere should be smaller than this but of the same order. 972π is approximately 972 * 3.14 ≈ 3051, which is a bit more than half of 5832, a plausible ratio for an inscribed sphere inside a cube. This confirms that the magnitude is reasonable.

Why Other Options Are Wrong:
11664π is 12 times larger than 972π and would correspond to using a radius larger than the cube edge, which is impossible. 36π and 288π are far too small and come from miscomputing r^3 or using radius instead of radius cubed. 432π corresponds to using an incorrect factor, such as (4/3)*r^2 instead of r^3. Only 972π matches the exact substitution into the correct formula.

Common Pitfalls:
A frequent mistake is to assume that the radius equals the cube edge instead of half of it. Another common error is to forget the factor (4/3) in the sphere volume formula or to compute 9^2 instead of 9^3. Keeping the formula V = (4/3)*pi*r^3 clearly in mind and computing step by step prevents these errors.

Final Answer:
972π