Longest rod inside a cube (space diagonal): What is the longest rod that can fit in a cubical vessel with side 10 cm?

Introduction / Context:For a cube, the longest segment entirely inside is the space diagonal, not the face diagonal or the edge. This checks formula recall and recognition of 3D geometry basics.

Given Data / Assumptions:

• Cubic side s = 10 cm
• Space diagonal formula: d = s√3

Concept / Approach:Know three canonical lengths: edge (s), face diagonal (s√2), and space diagonal (s√3). The rod must align with the space diagonal.

Step-by-Step Solution:d = s√3 = 10√3 cm

Verification / Alternative check:Construct via 3D Pythagoras: face diagonal = s√2; then √((s√2)^2 + s^2) = √(2s^2 + s^2) = s√3, confirming the formula.

Why Other Options Are Wrong:10 cm (edge) and 10√2 (face diagonal) are shorter than the space diagonal; “None” is false since 10√3 is listed; 5√6 equals 10√(1.5), still smaller than 10√3.

Common Pitfalls:Confusing face diagonal with space diagonal; forgetting the √3 factor for cubes.

Final Answer:10√3