In a certain rectangular prism, the total length of all the edges is 40 and the total surface area is 48. Find the length of the diagonal connecting one corner to the opposite corner.
+1223 Let the width be W, the length be L, and the height be H.
total length of all edges = 4(W + L + H) = 40 --> W + L + H = 10
total surface area = 2(WL + WH + LH) = 48 --> WL + WH + LH = 24
We want to find sqrt(W^2 + L^2 + H^2)
We can use the following algebraic identity:
(W + L + H)^2 = W^2 + L^2 + H^2 + 2WL + 2WH + LH
100 = W^2 + L^2 + H^2 + 2(24)
W^2 + L^2 + H^2 = 52
The space diagonal's length is therefore \(\boxed{\sqrt{52}}.\)
