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. \(10\sqrt{7}\) cm

Answer 1

The Correct Option is C

The problem at hand is to find the maximum length of a pencil that can be kept in a rectangular box with the dimensions 8 cm x 6 cm x 2 cm. To solve this, we need to compute the length of the diagonal of the box, as it represents the longest possible distance inside it.

The formula to calculate the diagonal \(d\) of a rectangular box with dimensions \(l\), \(w\), and \(h\) is given by:

\[ d = \sqrt{l^2 + w^2 + h^2} \]

Given:

• Length (\(l\)) = 8 cm
• Width (\(w\)) = 6 cm
• Height (\(h\)) = 2 cm

Substitute these values into the formula:

\[ d = \sqrt{8^2 + 6^2 + 2^2} \]

\[ d = \sqrt{64 + 36 + 4} \]

\[ d = \sqrt{104} \]

The diagonal length is \(\sqrt{104}\), which can be simplified as:

\[ d = \sqrt{4 \times 26} \]

\[ d = \sqrt{4} \times \sqrt{26} \]

\[ d = 2\sqrt{26} \]

Therefore, the maximum length of the pencil that can be kept in the box is \(2\sqrt{26}\) cm, which corresponds to the correct option given.