A rectangular box measures 10 cm × 8 cm × 5 cm. What is the maximum length of a pencil that can fit inside it (i.e., the space diagonal)?

Introduction / Context:The longest segment inside a rectangular box is its space diagonal. For dimensions a, b, c, the space diagonal d satisfies d^2 = a^2 + b^2 + c^2. Substitute the given side lengths and simplify the radical if possible.

Given Data / Assumptions:

• a = 10 cm, b = 8 cm, c = 5 cm.
• Space diagonal formula: d = √(a^2 + b^2 + c^2).

Concept / Approach:Compute the squared sum and factor to simplify.

Step-by-Step Solution:d^2 = 10^2 + 8^2 + 5^2 = 100 + 64 + 25 = 189d = √189 = √(9 * 21) = 3√21 cm

Verification / Alternative check:N/A beyond arithmetic; 3√21 ≈ 13.747 cm makes sense for a box with maximum side 10 cm.

Why Other Options Are Wrong:√150 and √98 correspond to smaller squared sums; 3√52 is too large (≈ 21.6 cm) and impossible for given box dimensions.

Common Pitfalls:Using face-diagonal formula √(a^2 + b^2) instead of space diagonal; forgetting to include all three squared terms.

Final Answer:3√21 cm