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Question

If \({A}_{1},{A}_{2}\) and \({A}_{3}\) denote the areas of three adjacent faces of a cuboid, then its volume is
(A) \({A}_{1}{A}_{2}{A}_{3}\)
(B) \(2{A}_{1}{A}_{2}{A}_{3}\)
(C) \(\sqrt{{A}_{1}{A}_{2}{A}_{3}}\)
(D) \(\sqrt [ 3 ]{ { A }_{ 1 }{ A }_{ 2 }{ A }_{ 3 } } \)

Answer

Answer: C
It is given that, \(A_,A_2,A_3\) be the areas of \(3\) adjacent faces of cuboid
Let \(V\) be the volume of cuboid.
Let dimensions of cuboid \(=l\times b\times h\)
\(A_1=l\times b\)
\(A_2=b\times h\)
\(A_3=h\times l\)
\(\Rightarrow\) \(V=l\times b\times h\)
Now,
\(\Rightarrow\) \(A_1A_2A_3=lb\times bh\times hl\)
\(\Rightarrow\) \(A_1A_2A_3=l^2b^2d^2\)
\(\Rightarrow\) \(A_1A_2A_3=(lbh)^2\)
\(\therefore\) \(A_1A_2A_3=V^2\)
\(\therefore\) \(V=\sqrt{A_1A_2A_3}\)
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