smc

We can create three equations using the given information. $$4a+4b+4c=13$$ $$2ab+2ac+2bc=\frac{11}{2}$$ $$abc=\frac{1}{2}$$ We also know that we want $\sqrt{a^2+b^2+c^2}$ because that is the length that can be found from using the Pythagorean Theorem. We cleverly notice that $a^2+b^2+c^2=(a+b+c{)}^2-2(ab+ac+bc)$. We know that $a+b+c=\frac{13}{4}$ and $2(ab+ac+bc)=\frac{11}{2}$, so $a^2+b^2+c^2={\left(\frac{13}{4}\right)}^2-\frac{11}{2}=\frac{169-88}{16}=\frac{81}{16}$. So our answer is $\sqrt{\frac{81}{16}}=\boxed{\frac{9}{4}}$. Interestingly, we don't use the fact that the volume is $\frac{1}{2}$.