The maximum length of a pencil that can be kept in a rectangular box of dimensions `8 cm xx 6 cm xx 2 cm` is ----

Options

A. `2 sqrt (13) cm`
B. `2 sqrt (14) cm`
C. `2 sqrt (26) cm`
D. `10 sqrt (2) cm`

Answer

To find the maximum length of a pencil that can be kept in a rectangular box with dimensions 8 cm, 6 cm, and 2 cm, we need to calculate the diagonal of the box. The formula for the diagonal \(d\) of a rectangular box is given by:

\[ d = \sqrt{L^2 + B^2 + H^2} \]

where:
- \(L\) is the length of the box,
- \(B\) is the breadth of the box,
- \(H\) is the height of the box.

### Step-by-Step Solution:

1. **Identify the dimensions of the box:**
- Length \(L = 8 \, \text{cm}\)
- Breadth \(B = 6 \, \text{cm}\)
- Height \(H = 2 \, \text{cm}\)

2. **Substitute the dimensions into the formula:**
\[ d = \sqrt{8^2 + 6^2 + 2^2} \]

3. **Calculate the squares of the dimensions:**
- \(8^2 = 64\)
- \(6^2 = 36\)
- \(2^2 = 4\)

4. **Add the squares together:**
\[ 64 + 36 + 4 = 104 \]

5. **Take the square root of the sum:**
\[ d = \sqrt{104} \]

6. **Simplify the square root:**
\[ \sqrt{104} = \sqrt{4 \times 26} = 2\sqrt{26} \]

7. **Final answer:**
The maximum length of the pencil that can be kept in the box is \(2\sqrt{26} \, \text{cm}\).