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) \(\sqrt [ 3 ]{ { A }_{ 1 }{ A }_{ 2 }{ A }_{ 3 } } \)

(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: C

It is given that, \(A_,A_2,A_3\) be the areas of \(3\) adjacent faces of cuboid

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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