SOLUTION: the sum of length, breadth and depth of a cuboid is 19 cm and the length of diagonal is 11 cm. find the surface area of the cuboid.

Question 1067836: the sum of length, breadth and depth of a cuboid is 19 cm and the length of diagonal is 11 cm. find the surface area of the cuboid.
Found 3 solutions by ikleyn, Alan3354, KMST:
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
The dimensions are 7 cm, 6 cm, 6 cm.

The surface area is 2*(7*6+7*6+6*6) = 240 cm^2.

Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
d^2 = L^2 + W^2 + H^2 = 11^2 = 121
L^2 + W^2 + H^2 = 121
L + W + H = 19
---------
--> 6 by 6 by 7
SA = 2LW + 2LH + 2WH
SA = 72 + 84 + 84 = 240 sq cm
Other solutions are possible.
-----
After thinking about it, that's the only solution.
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
Let , and be the length, breadth and depth of the cuboid.
and
--->
are out clues,
and of course
--->
--->

and yes, there are other options beyond {x,y,z} = {6,6,7} ,
but none with three integers, and the options fall in a narrow range.around
You can make ,
substitute into
to get an equation of degree 2 in and .
For the equation to have a solution for ,
you need .
To figure out if there would be another solution with integers,
I searched on line for "Pythagorean quadruples",
sets of 4 whole numbers, ,
such that .
The lists are given with the numbers
in such order that ,
so you only have to the first few
to realize that the only two ending in 11 are
2, 6, 9, 11 and 6, 6, 7, 11 .
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