Maybe there is a piece of this question I am missing. I could arrange the 56 pieces of 3" x 2.5" in any way I wanted and say that you would need a piece of steel those dimensions to avoid any waste at all. Does the large sheet of steel come in a particular shape?

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OK. I hope this wasn't a math homework problem or something because I can't really describe the process I used other than creative trial and error, but if you cut 28 squares from 2 larger sheets of steel 6" x 36" in size, you will only have 12 in.^{2} of waste at the end of all the cutting. I couldn't really find a one-sheet solution with the dimensions you gave that didn't waste a whole lot of steel, but if you do it this way, I think you'll be alright. I'll try to explain the layout:

Picture two rows of these tinier pieces with their 3" edges lined up against the 6" width of the larger sheet. With this layout, you can fit 14 of these tinier pieces (with their 2.5" dimension) into 35" - just 1" short of the 36" length of the larger sheet. So, out of one sheet, you've got half of your tiny pieces with just 6 in.^{2} of waste. Do this with another large sheet of steel with the same dimensions, and you'll have the other half of your pieces with only another 6 in.^{2} of waste for a total of only 12 in.^{2} of waste.

The total amount needed would be 56 X 3 X 2.5 Sq inch =420 sq in =12" X 35"

So the minimum size of the metal sheet required is 1ft X 3 ft = 3 Sq ft

You can easily cut out 14 strips of 1 ft length. Each of the strips can be cut to make 4 of 3" pieces, thus making a total of 4x 14= 56 pieces of 3"x 2.5"

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You have one rectangular sheet of steel, that must be cut into 56 pieces, therefore 56 is your total area (A). The area of a rectangle is Length x Width = A, so L x W = 56. What numbers can be multiplied to equal 56?

For example, 8 and 7, 8 x 7 = 56. This means your piece of steel is a rectangle with 8 pieces on one side (L), and 7 on the other (W).

Each of those individual pieces of steel, all 56 have the dimensions 3" by 2.5". Therefore to find the total dimension of the steel rectangle, you must multiply your sides by your L and W dimensions. 3" x 8 = 24" and 2.5" x 7 = 17.5". You need a sheet of steel that is 24" by 17.5".

But, keep in mind how you determined length and width, any numbers that multiply into 56 may be used.

I think the specified sizes she's got there in her comment have something to do with the problem, though. If we could solve the problem this way, we could say any dimensions we wanted and come up with a piece of steel for it (e.g. - 3" x 140" for a single row of 56 rectangles). There has to be some other stipulation for which the "least amount of waste" is significant.

## Comments

Maybe there is a piece of this question I am missing. I could arrange the 56 pieces of 3" x 2.5" in any way I wanted and say that you would need a piece of steel those dimensions to avoid any waste at all. Does the large sheet of steel come in a particular shape?

maybe one of these sizes?

Width

4"

12" (1 ft.)

24" (2 ft.)

36" (3 ft.)

Length

6"

12" (1 ft.)

24" (2 ft.)

36" (3 ft.)

48" (4 ft