The Rope of Dreams Recut: Polynomials of the Third Order – Cubic Equations

The Rope of Dreams in 3D

The Rope of Dreams in 3D

The Rope of Dreams Recut – Cubic Equations 

Did you make the most of The Rope of Dreams in my previous post “The Rope of Dreams : Polynomials of the Second Order – Quadratic Equations“. I hope so. Well now you have a chance to take your 120 meter rope and enter another Dimension with Cubic Equations.

What is the Scenario?

Once again you are given the Rope of Dreams, and a Golden Scissors which is the only thing that can cut the Rope of Dreams. You can cut the Rope of Dreams twice (cross section, no longitudinal cuts), which will give you 3 lengths. These lengths will be laid out one for each dimension X, Y, Z (i.e. left-right, backward-forward, up-down), and whatever volume you enclose anywhere on Earth is yours to keep, or do with whatever you wish. You can enclose only a single volume, a single time, and then must return the rope and the scissors. What would you do? You may not think of a Polynomial of the Third Order – a Cubic Equation, but you probably should. In this post we’re talking about cuboids, we’ll leave spherical shapes aside for now.

Do the Math!

This time we’ll do the mathematics first and then apply our findings to determine how best to maximise the volume we enclose. In the previous post we derived and proved the following quadratic equation:

Previously: \({ x^2 = (x-n)(x+n) + n^2 }\)

Multiply by \(x\): \({ x^3 = x ((x-n)(x+n) + n^2)}\)

Multiplying this out: \(\boxed{ x^3 =  (x-n) x (x+n) + xn^2}\)

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