Surface Area of Blocks Glued TogetherDate: 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 |
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