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The volume of a rectangular parallelepiped is given as 144 cc. Its surface area is given as 192 sq cm. And its corner to corner diagonal is given as 13 cm. How do I find out the three sides.

I have assumed the sides to be $a,b,c$. Now, $a^2+b^2+c^2=169, 2(ab+bc+ca)=192$, and $abc=144$. How do I solve the three equations without forming a cubic equation? (This is a class 9 problem so i am not suppoesd to use solution to cubic equations.)

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    Whenever I see a right angle and $13$ together, I think about $5$ and $12$. And when I see a right angle and $5$, I think about $3$ and $4$. Bingo.2012-08-25

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The quickest way is just to reacall pythagorean triples $(3,4,5)$ and $(5,12,13)$ due to 5 being common to both triples and 13 being the diagonal length thus establishing that 3, 4,and 12 satisfy $3^2+4^2+12^2=13^2$ as desired for the correct diagonal length. Notice that the product of these three numbers in the triples is 144 and they give the correct surface area.

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It seems likely that you are expected to use the "guess and check" procedure.

The product is $144$, not too many integer possibilities for $a$, $b$, $c$. Without loss of generality you may assume $a \ge b\ge c$. Also, the sum of squares condition tells you that $a \le 12$.

Why integers? Because the problem is meant to be solved easily by guess and check.

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    @SN77: The approach through the cubic is certainly more general. But in the question and comments, using the cubic was explicitly ruled out.2012-08-25