A rectangular parallelepiped has edges with integral lengths *x*, *y*, and *z*. The sum of the lengths of all twelve edges is 72 inches. The sum of the areas of all 6 faces is 212 square inches. The volume of the solid is 144 cubic inches. Find the length in inches of a diagonal of this solid.

We want L = √

*x*^{2}+*y*^{2}+*z*^{2}We know

4(

2(

*x*+*y*+*z*) = 72 →*x*+*y*+*z*= 182(

*xy*+*yz*+*xz*) = 212*xyz*= 144.We can get all of these numbers together by considering (

*x*+*y*+*z*)^{2}.(

→

→

→

→

→ L = √

*x*+*y*+*z*)(*x*+*y*+*z*) = 18^{2}= 324→

*x*^{2}+*xy*+*xz*+*yx*+*y*^{2}+*yz*+*zx*+*zy*+*z*^{2}= 324→

*x*^{2}+*y*^{2}+*z*^{2}+ 2(*xy*+*yz*+*xz*) = 324→

*x*^{2}+*y*^{2}+*z*^{2}+ 212 = 324→

*x*^{2}+*y*^{2}+*z*^{2}= 112→ L = √

*x*^{2}+*y*^{2}+*z*^{2}= √112 = √16 · 7 = 4√7