Answer: The maximum length of a toothpick that still fits the box is [tex]7\sqrt{2} \,\,\mbox{cm}[/tex]
Explanation: the longest straight distance within a rectangular box (prism) is its diagonal (in three dimensions), i.e., the path from one corner to its diagonally opposite corner.
The diagonal length in a two-dimensional rectangle can be calculated using the Pythagorean Theorem. There is a three-dimensional analog of that theorem for prisms. Let a,b,c be the sides of the prism. Then the following holds for the diagonal length d:
[tex]d^2=a^2+b^2+c^3[/tex]
In order to answer the question we determine the length d:
[tex]d = \sqrt{3^2+5^2+8^2}=\sqrt{2\cdot 49}=7\sqrt{2} \,\,\mbox{cm}[/tex]
