Surface Area of Blocks Glued Together

Date: 09/09/2001 at 23:15:33
From: Jodi Meyers
Subject: Trig?

Three cubes whose edges are 2, 6, and 8 centimeters long are glued 
together at their faces. Compute the minimum surface area possible for 
the resulting figure.

I have tried this so many ways that I am not sure where I started and 
where I have left off. Three cubes that are 2 cms long, 6 cms high, 
and 8 cms deep would be 3(2+6)+ = 6+18+24 cms in total area, which is 
not right. URGH. HELP. 

Regards, 
Jodi Meyers

Date: 09/10/2001 at 14:44:59
From: Doctor Ian
Subject: Re: Trig?

Hi Jodi,

There are two ways to interpret what you've written.  Strictly 
speaking, 

  three cubes whose edges are 2, 6, and 8 cm long

would mean three cubes, 

       2
     +--+
  2 /  /| 
   +--+ | 
 2 |  |/
   +--+

          6
      +------+
   6 /      /|
    /      / | 
   +------+  | 
   |      |  |
 6 |      |  /
   |      | /
   +------+

          8
       +--------+
      /        /|
   8 /        / |
    /        /  | 
   +--------+   | 
   |        |   |
 8 |        |   |
   |        |  /
   |        | /
   +--------+


Note that all the edges of a cube have to be the same length, or it's 
not a cube. I think that you're probably talking about 

  three rectangular prisms measuring 2 cm by 6 cm by 8 cm

i.e., 

      +---------+
   2 /         /|
    +---------+ |            [and two more just like it]
    |         | /     
  6 |         |/
    +---------+
         8

Is this correct?  If so, then the three blocks ('block' is easier to 
type than 'rectangular prism'), if not touching, would have a total 
surface area of 

  area = 3 * 2(2*6 + 2*8 + 6*8)

       = 6(12 + 16 + 48)

Do you see why?  The '3' is there because there are three blocks; each 
of the products in parentheses is the area of one side; and there are 
two sides of each size in a block. 

Anyway, so this is the maximum surface area. By gluing the blocks 
together, you can hide some of that area. For example, if you glue two 
blocks together on the 2 x 6 face,

      +---------+---------+
   2 /         /         /|
    +---------+---------+ | 
    |         |         | /
  6 |         |         |/
    +---------+---------+
         8        8

you've hidden two 2 x 6 faces, which means that you've eliminated 
2(2 * 6) square centimeters of surface area, leaving you with an area 
of
  
  area = total - hidden

       = 6(12 + 16 + 48) - 2(2 * 6)

So what you want to do is play around with the various ways that you 
can hide faces, or parts of faces. For example, if you glue two blocks 
together on the 6 x 8 face, you can hide 2(6 * 8) square centimeters 
of surface area.  

Note that as you do this, you only have to pay attention to the 
'hidden' part of the equation, since the 'total' part stays the same 
no matter how you're gluing the blocks together. 

Once you think you've found an optimal configuration, the final 
surface area will be the total area for the blocks minus the area that 
you've managed to hide. 

Can you take it from here? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 09/11/2001 at 00:25:26
From: Jodi Meyers
Subject: Re: Trig?

I bow before thee. Thank you, kind sir. :) 

Jodi Meyers