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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The Rope of Dreams : Polynomials of the Second Order – Quadratic Equations

Where would you go? What would you enclose?

Where would you go? What would you enclose?

 

 

The Rope of Dreams – Polynomial

Imagine you are given a length of rope that is 120 meters long, and told that you can go to any place on Earth and whatever you enclose with the rope – is yours to keep, or do with whatever you wish. You can enclose only a single area, a single time, and then must return the rope. What would you do? You may not think of a Polynomial of the Second Order – a Quadratic Equation, but you probably should.

What would you do?

Say a big “Thanks”, take the rope, and start pondering the options. A likely plan is to think of where on Earth you want to go (a tropical island, a bustling city, a countryside retreat, maybe even Fort Knox – it’s your choice), and while en route to your destination figure out how to maximize the area the 120 meter rope can enclose. I’ll leave the destination to your own imagination (you can post in the Comments section below) and turn our attention for now to maximizing the area the rope can enclose once you get there. Did someone say Polynomial!

How long is a piece of string, or 120 meters of rope?

A likely first question you might have is to get an idea of just how long 120 meters is, so some reference examples might help, note that ‘m’ is short for ‘meter’. A soccer pitch is between 90m and 120m in length; A rugby pitch is 100m – the same as the 100m sprint in Athletics (Usain Bolt, Carl Lewis etc.); An American football pitch is 110m long; A CLG/GAA (Cumann Lúthchleas Gael / Gaelic Athletic Association) pitch is between 130m – 145m in length. For petrol heads, 120m is about 24 Nascars end-to-end, or 21 Formula1 cars end-to-end – that’s almost the entire grid – are you heading to Monaco with your rope?

Triangles and Rectangles and Squares, Oh My!

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