A rectangular prism has a total surface area of 56. Also, the sum of all the edges of the prism is \(56\). Find the length of the diagonal joining one corner of the prism to the opposite corner.
+9519 So, let \(h \times w \times l\) be the dimensions of that prism.
We have \(\begin{cases}2(hw + wl + hl) = 56\\h + w + l = 56\end{cases}\). We are to find \(\sqrt{h^2 + w^2 + l^2}\).
We can use the identity \((h + w + l)^2 = (h^2 + w^2 + l^2) + 2(hw + wl + hl)\) to calculate \(h^2 + w^2 + l^2\), and then take square root. I will leave that as an exercise for you.
If you don't believe the identity is true, try to expand the left hand side and the right hand side. \((h + w + l)^2 = ((h + w) + l)^2 = (h + w)^2 + 2l(h + w) + l^2\) and etc.
