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# help

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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?

Jun 17, 2020

#1
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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 = $$\sqrt 3 : \sqrt 5$$

Jun 17, 2020