AMC12 2014 12B, Problem 14

A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?

Solution: Let the side lengths of the rectangular box be $x, y$  and $z$. From the information we get \[4(x+y+z) = 48 \Rightarrow x+y+z = 12\] \[2(xy+yz+xz) = 94\] The sum of all the  lengths of the box's interior diagonals is \[4 \sqrt{x^2+y^2+z^2}\] Squaring the first expression, we get: \[144 =(x+y+z)^2 =  x^2+y^2+z^2 + 2(xy+yz+xz)\] \[144 =  x^2+y^2+z^2 + 94\] Hence \[x^2+y^2+z^2 = 50\] \[4 \sqrt{x^2+y^2+z^2} =  \fbox{20\sqrt{2}}\]

Answer: 20\sqrt{2}