cinelli wrote:A cuboid is constructed so that
1) it has a volume of 7.5 cubic cms
2) its total surface area is 26 square cms
3) the total length of all its edges is 26 cms.
What are the cuboid’s dimensions?
I'm still rather uncertain about the value of spoiler separators on these boards, but one won't do any real harm...
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Let the three dimensions in centimetres of the cuboid be a, b and c.
Then:
volume = abc = 7.5
area of all faces = 2(ab+bc+ca) = 26, so ab+bc+ca = 13
length of all edges = 4(a+b+c) = 26, so a+b+c = 6.5.
Now consider the cubic equation:
(x-a)(x-b)(x-c) = 0
Clearly its three roots are a, b and c. But (x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab+bc+ca)x - abc = x^3 - 6.5x^2 + 13x - 7.5.
So the dimensions of the cuboid are the three roots of x^3 - 6.5x^2 + 13x - 7.5 = 0. So just apply the standard formulae for solving a cubic and there's the solution...
What? You don't know the standard formulae for solving a cubic??? Shame on you!
And shame on me as well - I don't know them either! I can look them up - but I never do, because they're messy and not very informative... When faced with a cubic equation, I either try to spot a solution - if I get one, I can factor it out and be left with a quadratic equation, which I do know the standard formula for - or I solve it numerically.
In this case, it's quite easy to spot that x=1 is a solution, since it causes x^3 - 6.5x^2 + 13x - 7.5 to be 1 - 6.5 + 13 - 7.5 = 0. Factoring it out, I get x^3 - 6.5x^2 + 13x - 7.5 = (x-1)(x^2-5.5x+7.5), so the other two dimensions are the roots of the quadratic equation x^2-5.5x+7.5 = 0, which are (5.5+/-sqrt(5.5^2-4*1*7.5))/2 = (5.5+/-sqrt(30.25-30))/2 = (5.5+/-0.5)/2 = 3 or 2.5.
So the dimensions of the cuboid are 1, 2.5 and 3.
Gengulphus