SOLUTION: The surface area and diagonal of a cuboid are 288 sq.cm and 12 cm respectively. Show that the cuboid is a cube.

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Question 1099031: The surface area and diagonal of a cuboid are 288 sq.cm and 12 cm respectively. Show that the cuboid is a cube.
Found 2 solutions by ankor@dixie-net.com, ikleyn:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
The surface area and diagonal of a cuboid are 288 sq.cm and 12 cm respectively.
Show that the cuboid is a cube.
:
let s = one side of the cube
therefore the surface area
6s^2 = 288
s^2 = 288/6
s^2 = 48
:
The diagonal of one side
d = sqrt%282s%5E2%29
The diagonal of the cuboid
sqrt%282s%5E2+%2B+s%5E2%29 = 12
square both side
2s%5E2+%2B+s%5E2 = 144
replace s^2 with 48
%282%2848%29%29+%2B+48%29 = 144
96 = 144 - 48
96=96

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
This condition means EXACTLY THE FOLLOWING: 

    We are given a rectangular box (a PRISM) with dimensions "a", "b" and "c".
    The surface area of the box is 288 sq. cm.
    The 3D diagonal of the box is 12 cm long.
    Prove that the box is a cube.


Therefore, the solution by the other tutor is IRRELEVANT.


The correct solution is THIS: 

    Let us consider the quadratic form 

        Q(a,b,c) = %28a-b%29%5E2+%2B+%28b-c%29%5E2+%2B+%28a-c%29%5E2.     (1)


     We have Q(a,b,c) = %28a%5E2+-+2ab+%2B+b%5E2%29 + %28b%5E2+-+2bc+%2B+c%5E2%29 + %28a%5E2+-+2ac+%2B+c%5E2%29 = 

                      = 2%2A%28a%5E2+%2B+b%5E2+%2B+c%5E2%29 - 2%2A%28ab+%2B+bc+%2B+ac%29.   (2)


     As everybody knows  (or MUST know),

        a%5E2+%2B+b%5E2+%2B+c%5E2 is the square of the length of the 3D diagonal, so it is equal to 12%5E2 cm^2.

     Therefore the term  2%2A%28a%5E2+%2B+b%5E2+%2B+c%5E2%29  in (2) is equal to  2%2A12%5E2 = 288 cm^2.



     Again, as everybody knows (or MUST know),

        2%2A%28ab+%2B+bc+%2B+ac%29  is the surface area of the rectangular prism.

     Therefore,  the term  2%2A%28ab+%2B+bc+%2B+ac%29  in  (2)  has the given value of 288 cm^2 under the given condition.



     Now it is clear that at given data the form  Q(a,b,c)  is equal to zero.

     But  Q(a,b,c) is the sum of squares, and can be equal to zero if and only if a = b,  b = c  and a = c.



     It implies that  a = b = c  and the prism is a cube under given condition.

QED.


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