The Dumbells Soup Company makes its own cylindrical tin cans. The cans have a diameter of 3 inches and they are 3.5 inches tall.

Dumbells produces the cans by cutting out circles and rectangles from a large sheet of tin. The "wasted tin" between circles and rectangles that they cut out is thrown away. The company can order sheets of arbitrary length, but they are always 6 feet wide.

The operations manager just received an order for 100 cans. He should order the shortest length sheet of tin he can, because he is tasked with minimizing the wasted tin.

What length should the manager order?

I didn't get real enthusiastic about this problem too, but the flurry of activity over it is frightening, so, gosh, why not? Okay, and, like, mind that I am making this up as I go along ... lol ... I need a calculator. Okay, like, got one.

Group all the rectangles. Put 20 x 3.5" along the 6 foot width which means 5 x 3pi columns or 15pi length to make 100. That's 47.1238898 length x 70" width ... 2" waste at the top. Nope?

Turn the rectangles, that's 3pi or 9.424777961" along the width which allows a column of 7 and is 65.97344573 wide, leaving 6.026554274" at the top. With 7, I need 15 x 3.5" or 52.5" long which will allow 100 rectangles plus room for 5 more ... put circles in the leftover there and at the top.

So, 34 circles at the top and 15 where there's room for 5 more rectangles. So that's 49 of the 200 circles. I need 151 more. Now I put 24 circles along the width next to the rectangles and then 23 nestled to them and another 24 and so on. That's 7 columns with the centers on triangles and vertical or horizontal spacing of 2.598076211 x 6 = 15.58845727" + 3" = 18.58845727 ... room for 165 circles and I only need 151 ... more waste.

Anywhatever, 52.5 + 18.58845727 = 71.08845727"

However, arranging everything like at the start with the 2" waste at the top, and using 9 columns of circles with the centers on triangles adds to 70.90849946" ... I'm not typing all that into here ... so, the first way is better ...

The way I figure it, the manager orders a 70.90849946" length of 6 foot wide sheet steel.

Ayyyannnd ... this is picky nit silly. What a trip! Still, it's a good problem. How much is their mathematician costing them per hour? Lol oh lol.

Posted by CeeAnne on 2004-11-13 03:32:24