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

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 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

What happens if we look at a cuboid instead of a cube? Can we get the missing orders now?

Let’s start with a cuboid with sides x, x, and y.

plotting z = x2y, z = 4xy + 2x2, z = 8x + 4y. V = x2y S = 4xy + 2x2 E = 8x + 4y So we can work in 3D, plotting z = x2y, z = 4xy + 2x2, z = 8x + 4y. Autograph File 2

E < S < V V < E < S E < V < S V < S < E It seems we can manage these orders, but no others: Red < Green < Purple Purple < Red < Green Red < Purple < Green Purple < Green < Red E < S < V V < E < S E < V < S V < S < E So we get the same orders that we had with the cube...

There’s another way to look at this: Take a cuboid with sides x, x, and kx What happens as we vary k? Autograph File 3

Or, taking logs with y = (8+4k)x, y = (2+4k)x2,y = kx3 gives us log y = logx + log (8+4k) logy = 2logx + log (2+4k) log y = 3logx + log k and now we can plot log y v log x, And we have three straight lines as before, And only four possible orders. So no new orders are possible!

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

then (4x+4y+4z)xyz > (2xy+2yz+2zx)2 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.

What happens if we vary V, S and E? Just the one. Autograph File 4

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