www.carom-maths.co.uk Activity 2-8: V, S and E

Do you have access to Autograph? If you do, then clicking on the links in this Powerpoint should open Autograph files automatically for you. But if you don’t.... Click below, and you will taken to a file where Autograph is embedded. Autograph Activity link http://autograph-maths.com/ activities/maxhikorski/carom-2-12-1.html

E = total edge length S = total surface area V = volume

There are six ways to write E, S and V in order of size. Interesting question: can you find a cube for each order? If not, what about a cuboid?

Let’s try a cube, of side x: E = 12x, S = 6x2, V = x3 We can plot y = 12x, y = 6x2, y = x3 together… Autograph File 1 http://www.s253053503.websitehome.co.uk/carom/carom-files/carom-2-12-3.agg Only four regions!

log y = logx + log 12 logy = 2logx + log 6 log y = 3logx Or, taking logs with y = 12x, y= 6x2,y = x3 gives us log y = logx + log 12 logy = 2logx + log 6 log y = 3logx and now we can plot log y v log x:

It’s clear that only 4 out of 6 orders are possible.

0 < x < 2 V < S < E The four possible orders are: 0 < x < 2 V < S < E 2 < x < √12 V < E < S √12 < x < 6 E < V < S 6 < x E < S < V

Can we find a cuboid with sides x, y, z such that S < E and S < V?

We need; xyz > 2xy + 2yz + 2zx and 4x + 4y + 4z > 2xy + 2yz + 2zx Now if a > b > 0 and c > d > 0, then ac > bd > 0

4x2yz+4xy2z+4xyz2 > 4x2yz+4xy2z+4xyz2+f(x, y, z) So if xyz > 2xy + 2yz + 2zx > 0 and 4x + 4y + 4z > 2xy + 2yz + 2zx > 0 then (4x+4y+4z)xyz > (2xy+2yz+2zx)2 So 4x2yz+4xy2z+4xyz2 > 4x2yz+4xy2z+4xyz2+f(x, y, z) where f(x, y, z) > 0. Contradiction!

If x = 3, y = 4 and z = 5, then V = 60, S = 94, E = 48. Is there another cuboid where the values for V, S, and E are some other permutation of 60, 94 and 48?

(2x-a)(2x-b)(2x-c) = 8x3 – Ex2 + Sx – V = 8x3 - 4(a+b+c)x2 + 2(ab+bc+ca)x - abc = 8x3 – Ex2 + Sx – V where E, S and V are for the cuboid with sides a, b and c. The equation 8x3 – Ex2 + Sx – V = 0 has roots a/2, b/2 and c/2.

So our question becomes: which of the following six curves has three positive roots? y = 8x3  48x2 + 94x – 60, y = 8x3  48x2 + 60x – 94, y = 8x3  94x2 + 48x – 60, y = 8x3  94x2 + 60x – 48, y = 8x3  60x2 + 94x – 48, y = 8x3  60x2 + 48x – 94.

Just the one. What happens if we vary V, S and E? Autograph File 2 http://www.s253053503.websitehome.co.uk/carom/carom-files/carom-2-12-4.agg

a = 8, b = 30, c = 29. Yellow: roots are 0.4123..., 1.2127..., 2, sides are double. V = 8, S = 30, E = 29. Green: roots are 0.5, 0.8246..., 2.4254..., sides are double. V = 8, S = 29, E = 30.

Carom is written by Jonny Griffiths, hello@jonny-griffiths.net With thanks to: Rachel Bolton, for posing the interesting question at the start. Douglas Butler. Carom is written by Jonny Griffiths, hello@jonny-griffiths.net