brute force method to choose the answer?

Question

The sum of the perimeters of each of the six faces of a cuboid is 64 cm and the total surface area of the cuboid is 10 cm^2

If the length of the cuboid is greater then its height,what could be the greatest possible value(in cm) of its length ?

1)20/3
2)22/3
3)sqrt(54)
4)sqrt(55)

which answer to choose ?

any brute force method for quick choosing ?

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seashells .....cuboid example is brick

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No..i need a brute force method to choose the answer.

i dont need the complete solution.....because lack of time.
some alternative solution.

thats why i specifically mentioned a brute force method to solve this problem

Answer 1

Call the 3 lengths of the cuboid l1, l2 and l3.

Two faces have sides l1 and l2
Two have l2 and l3
Two have l1 and l3.

So 4*( l1+l2)+4*(l1+l3)+4(l2+l3)=64. l1+l2+l3=8
l1*l2 +l2*l3 + l3*l1 =5

x+y+z=8
xy + yz +xz =5

First let's arrange your possible solutions from the highest to the lowest.

The order is
1) sqrt(55)
2) sqrt(54)
3) 22/3
4) 20/3

Now let's try them in that order and see if they can be true or not.

z= sqrt(55) then:

x+y = 8 - 7.416 = 0.584
xy +7.416*0.584 = 5
xy=0.669

Does this work? x(0.584 - x)=0.669

x^2 - 0.584 x + 0.669 = 0

Try this, this is as "brutish" as I can get.