If the rectangular faces of a brick have their diagonals in the ratio 3 : 2 \sqrt{3} : \sqrt{15}, what is the ratio of the length of the shortest edge of the brick to that of its longest edge?
+9519 Let a, b, c be the edges.
\(\begin{cases}ab = 3\\bc = 2\sqrt 3\\ca = \sqrt{15}\end{cases}\)
(1) * (3) / (2) :
\(a^2 = \dfrac{3\sqrt{15}}{2\sqrt 3} = \dfrac{3\sqrt 5}2\\ a = \sqrt{\dfrac{3\sqrt 5}2}\)
Similarly,
\(b = \sqrt{\dfrac6{5}\sqrt5}\) and \(c = \sqrt{2\sqrt 5}\)
Required ratio = b : c = \(\sqrt 3 : \sqrt 5\)
