A cube has volume 729 cubic centimetres. Find the length of its space diagonal (in centimetres).

Introduction / Context:The space diagonal of a cube connects opposite vertices and equals a√3, where a is the edge length. If the volume is known, we compute a from V = a^3 and then form the space diagonal directly. This problem tests chaining two standard cube identities correctly.

Given Data / Assumptions:

• Volume V = 729 cm^3.
• Cube identities: V = a^3 and space diagonal d = a√3.

Concept / Approach:Find a = ∛V. Then substitute a into d = a√3. Keep radicals exact; the number is crafted so the cube root is integral.

Step-by-Step Solution:a = ∛729 = 9 cmd = a√3 = 9√3 cm

Verification / Alternative check:Confirm volume from a = 9: 9^3 = 729 cm^3; thus the diagonal 9√3 cm is consistent.

Why Other Options Are Wrong:9√2 cm is the face diagonal for a = 9 (a√2), not the space diagonal; 18 cm is twice the edge, not applicable; 18√3 omits the division by 2 that would appear in some other contexts, not here.

Common Pitfalls:Using face diagonal a√2 instead of space diagonal a√3; miscomputing the cube root of 729.

Final Answer:9√3 cm