## Cube Volumes

Posted by Karl Sharman on June 17, 2010 – 12:13 pm

For any given cuboid it is possible to measure up to three different perimeters. Given that cuboid A has perimeters 24, 32, and 40, and cuboid B has perimeters 24, 32, and 48, which cuboid has the greatest volume?

Anyone answering “A” or “B” without showing their working out will be deemed to be guessing. No points for guessing!

Karl Sharman

June 17th, 2010 at 1:30 pm

Cuboid A:

perimeter 24 consists of sides with lengths 8,4,8,4

perimeter 32 consists of sides with lengths 12,4,12,4

perimeter 40 consists of sides with lengths 8,12,8,12

Volume = L * W * H

= 12 * 8 * 4

= 384

Cuboid A:

perimeter 24 consists of sides with lengths 10,2,10,2

perimeter 32 consists of sides with lengths 2,14,2,14

perimeter 48 consists of sides with lengths 10,14,10,14

Volume = L * W * H

= 14 * 10 * 2

= 280

Therefore Cuboid A has greatest volume.

Cube has 8 corners and 12 segments. Suppose cube has corners on the front face labelled A, B, C, & D from upper left corner and E, F, G, & H on back face from upper left corner (as viewed from the front)

Cube A Lengths of each segment:

Front Face

AB 8

BC 4

CD 8

DA 4

Left Side

AE 12

EH 4

HD 12

DA 4

Top Side

AE 12

EF 8

FB 12

BA 8

Cube A Lengths of each segment:

Front Face

AB 10

BC 2

CD 10

DA 2

Left Side

AE 14

EH 2

HD 14

DA 2

Top Side

AE 14

EF 10

FB 14

BA 10

June 17th, 2010 at 1:50 pm

A < B

24 + 32 + 40 < 24 + 32 + 48

June 17th, 2010 at 1:55 pm

cuboid A’s dimensions are 4, 8 and 12:

2*4+2*8=24, 2*4+2*12=32 and 2*8+2*12=40

it’s easy to calculate this, because you have the three equations a+b=24/2, a+c=32/2 and b+c=40/2

(or just use trial and error)

so cuboid A’s volume is 4*8*12=384

cuboid B’s dimensions are 2, 10 and 14:

2*2+2*10=24, 2*2+2*10=32 and 2*10+2*14=48

so cuboid B’s volume is 2*10*14=280

thus A’s volume is greater than B’s volume

June 19th, 2010 at 3:53 pm

Well done John24 and Carc. Obviously too easy….

Still, an intersting problem as the obvious response would be that B’s volume is the greater with the gretaer measurements, but not the case as you have proved.

June 23rd, 2010 at 1:37 pm

A cube has 6 sides