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. `2vv 13 cm `
B. 2^14 cm
C. 2-^ 26 cm
D. `10 vv 2cm `

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 diagonal represents the longest straight line that can fit inside the box.

### Step-by-Step Solution:

1. **Identify the dimensions of the box**:
- Length (l) = 8 cm
- Width (w) = 6 cm
- Height (h) = 2 cm

2. **Use the formula for the diagonal of a cuboid**:
The formula to calculate the diagonal (d) of a rectangular box (cuboid) is given by:
\[ d = \sqrt{l^2 + w^2 + h^2} \]

3. **Substitute the dimensions into the formula**:
\[ d = \sqrt{(8 \, \text{cm})^2 + (6 \, \text{cm})^2 + (2 \, \text{cm})^2} \]

4. **Calculate the squares of the dimensions**:
- \( (8 \, \text{cm})^2 = 64 \, \text{cm}^2 \)
- \( (6 \, \text{cm})^2 = 36 \, \text{cm}^2 \)
- \( (2 \, \text{cm})^2 = 4 \, \text{cm}^2 \)

5. **Add the squared values**:
\[ 64 \, \text{cm}^2 + 36 \, \text{cm}^2 + 4 \, \text{cm}^2 = 104 \, \text{cm}^2 \]

6. **Take the square root of the sum**:
\[ d = \sqrt{104 \, \text{cm}^2} \]

7. **Simplify the square root**:
- Factor 104: \( 104 = 4 \times 26 \)
- Thus, \( \sqrt{104} = \sqrt{4 \times 26} = \sqrt{4} \times \sqrt{26} = 2\sqrt{26} \)

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