A right rectangular prism (storage container) has inner dimensions $5\text{ m} \times 4\text{ m} \times 3\text{ m}$. The length of the longest straight pipe in metres that can fit completely inside the container is expressed as $\sqrt{k}$. The value of integer $k$ is ________.
Answer Key:
50
Step-by-Step Solution & Explanation
1. The longest straight segment in a rectangular cuboid of length $L$, width $W$, and height $H$ is the 3D space diagonal:
$D = \sqrt{L^2 + W^2 + H^2}$
2. Substituting the dimensions:
$D = \sqrt{5^2 + 4^2 + 3^2} = \sqrt{25 + 16 + 9} = \sqrt{50}\text{ m}$
3. Therefore, $k = 50$.