Question

The maximum length of a pencil that can be kept in a rectangular box of dimensions 8 cm x 6 cm x 2 cm, is

• A. \(2\sqrt{13}\) cm
• B. \(2\sqrt{14}\) cm
• C. \(2\sqrt{26}\) cm
• D. \(10\sqrt{2}\) cm

Hint:

The longest object that fits in a box lies along its space diagonal, found using the 3D Pythagoras formula.

Answer 1

The Correct Option is C

Step 1: Identify what maximum length means for a box.
The longest straight object that fits inside a rectangular box lies along its main space diagonal, the line joining one corner to the farthest opposite corner. For a box of length \(l\), breadth \(b\) and height \(h\), this diagonal is \(\sqrt{l^2+b^2+h^2}\).

Step 2: Plug in the given dimensions.
Here \(l = 8\), \(b = 6\) and \(h = 2\), so the diagonal \( = \sqrt{8^2+6^2+2^2} = \sqrt{64+36+4} = \sqrt{104}\).

Step 3: Simplify the surd.
\(\sqrt{104} = \sqrt{4 \times 26} = 2\sqrt{26}\).

Final Answer:
The maximum pencil length that fits in the box is \(2\sqrt{26}\) cm, so option C is correct. \[ \boxed{2\sqrt{26} \text{ cm}} \]