+273 A rectangular prism has a total surface area of 56. Also, the sum of all the edges of the prism is 64. Find the length of the diagonal joining one corner of the prism to the opposite corner.
+2668 The length of the diagonal from one corner to the opposite is just \(\sqrt{x^2 + y^2 + z^2}\)
We have \(4(x + y + z)= 64\), meaning \(x + y + z = 16\)
We also know that \(2(xy + xz + yz) = 56\).
Note that\((x+y+z)^2 = x^2+y^2+z^2 + 2xy + 2xz + 2yz = 16^2 = 256\)
But, by the magic of substitution, we already know that \(2xy + 2yz + 2xz = 56\). Subbing this in gives us \(x ^2 + y^2 + z^2 = 200\).
So, the length of the diagonal is \(\sqrt{x^2 + y^2 + z^2} = \sqrt{200} = \sqrt{100} \times \sqrt{2} = \color{brown}\boxed{10 \sqrt 2}\)
+2668 The length of the diagonal from one corner to the opposite is just \(\sqrt{x^2 + y^2 + z^2}\)
We have \(4(x + y + z)= 64\), meaning \(x + y + z = 16\)
We also know that \(2(xy + xz + yz) = 56\).
Note that\((x+y+z)^2 = x^2+y^2+z^2 + 2xy + 2xz + 2yz = 16^2 = 256\)
But, by the magic of substitution, we already know that \(2xy + 2yz + 2xz = 56\). Subbing this in gives us \(x ^2 + y^2 + z^2 = 200\).
So, the length of the diagonal is \(\sqrt{x^2 + y^2 + z^2} = \sqrt{200} = \sqrt{100} \times \sqrt{2} = \color{brown}\boxed{10 \sqrt 2}\)
