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 = √x2 + y2 + z2
We know
4(x + y + z) = 72 → x + y + z = 18
2(xy + yz + xz) = 212
xyz = 144.
2(xy + yz + xz) = 212
xyz = 144.
We can get all of these numbers together by considering (x + y + z)2.
(x + y + z)(x + y + z) = 182 = 324
→ x2 + xy + xz + yx + y2 + yz + zx + zy + z2 = 324
→ x2 + y2 + z2 + 2(xy + yz + xz) = 324
→ x2 + y2 + z2 + 212 = 324
→ x2 + y2 + z2 = 112
→ L = √x2 + y2 + z2 = √112 = √16 · 7 = 4√7
→ x2 + xy + xz + yx + y2 + yz + zx + zy + z2 = 324
→ x2 + y2 + z2 + 2(xy + yz + xz) = 324
→ x2 + y2 + z2 + 212 = 324
→ x2 + y2 + z2 = 112
→ L = √x2 + y2 + z2 = √112 = √16 · 7 = 4√7