A rectangular box having length, breadth and height as $(a + b + c), (a^{2} + b^{2}) a n d (2 a + 2 b + 2 c)$ respectively. Find the volume of the box.

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Solution

Total volume =

$(a + b + c) (a^{2} + b^{2}) (2 a + 2 b + 2 c)$

$= (a^{3} + a b^{2} + a^{2} b + b^{3} + c a^{2} + c b^{2}) (2 a + 2 b + 2 c)$

$= (2 a^{4} + 2 a^{3} b + 2 a^{3} c + 2 a^{2} b^{2} + 2 a b^{3} + 2 a b^{2} c + 2 a^{3} b + 2 a^{2} b^{2} + 2 a^{2} b c + 2 a b^{3} + 2 b^{4} + 2 b^{3} c + 2 a^{3} c + 2 a^{2} b c + 2 a^{2} c^{2} + 2 a b^{2} c + 2 b^{3} c + 2 b^{2} c^{2})$

$= (2 a^{4} + 4 a^{3} b + 4 a^{3} c + 4 a^{2} b^{2} + 4 a b^{3} + 4 a b^{2} c + 4 a^{2} b c + 2 b^{4} + 4 b^{3} c + 2 a^{2} c^{2} + 2 b^{2} c^{2})$

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