Question 17
A rectangular box $\mathcal{P}$ has distinct edge lengths $a,$ $b,$ and $c.$ The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13,$ the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2},$ and the volume of $\mathcal{P}$ is $\frac{1}{2}.$ What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}?$