A) \[\frac{xyz}{2\sqrt{2}}\]
B) \[\frac{\sqrt{({{y}^{2}}+{{z}^{2}}-{{x}^{2}})({{z}^{2}}+{{x}^{2}}-{{y}^{2}})({{x}^{2}}+{{y}^{2}}-{{z}^{2}})}}{2\sqrt{2}}\]
C) \[\frac{\sqrt{({{y}^{2}}+{{z}^{2}})({{z}^{2}}+{{x}^{2}})({{x}^{2}}+{{y}^{2}})}}{2\sqrt{2}}\]
D) None of these
Correct Answer: B
Solution :
| Let l, b and A be the sides of cuboid. |
| \[{{l}^{2}}+{{b}^{2}}={{x}^{2}}\] ?(i) |
| \[{{b}^{2}}+{{h}^{2}}={{y}^{2}}\] ?(ii) |
| and \[{{h}^{2}}+{{l}^{2}}={{z}^{2}}\] ?(iii) |
| \[2\,\,({{l}^{2}}+{{b}^{2}}+{{h}^{2}})={{x}^{2}}+{{y}^{2}}+{{z}^{2}}\][From Eqs. (i), (ii) and (iii)] |
| \[\Rightarrow \]\[{{l}^{2}}+{{b}^{2}}+{{h}^{2}}=\frac{1}{2}({{x}^{2}}+{{y}^{2}}+{{z}^{2}})\] |
| From Eqs. (i), (ii), (iii) and (iv), we get |
| \[h=\sqrt{\frac{{{y}^{2}}+{{z}^{2}}-{{x}^{2}}}{2}},l=\sqrt{\frac{{{z}^{2}}+{{x}^{2}}-{{y}^{2}}}{2}}\]and |
| \[b=\sqrt{\frac{{{x}^{2}}+{{y}^{2}}-{{z}^{2}}}{2}}\] |
| Hence, volume of cuboid \[=lbh\] |
| \[=\sqrt{\frac{({{y}^{2}}+{{z}^{2}}-{{x}^{2}})({{z}^{2}}+{{x}^{2}}-{{y}^{2}})({{x}^{2}}+{{y}^{2}}-{{z}^{2}})}{2\times 2\times 2}}\] |
| \[=\frac{1}{2\sqrt{2}}\sqrt{({{y}^{2}}+{{z}^{2}}+{{x}^{2}})({{z}^{2}}+{{x}^{2}}-{{y}^{2}})({{x}^{2}}+{{y}^{2}}-{{z}^{2}})}\] |
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