My Math Forum rectangulars fit in one rectangular

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 October 8th, 2010, 12:30 AM #1 Newbie   Joined: Oct 2010 Posts: 2 Thanks: 0 rectangulars fit in one rectangular Dear, I'm trying to find to right equations for the following problem: I have one big rectangular with fixed dimensions (lenght = a ; width = b). I have 10 other rectangulars with variable dimensions (for example x1= lenght of first rectangle, y1 = width of first rectangle). What is the formula to see if the 10 rectangulars will fit in the big one. Note: all 10 may be turned and combined with the others. Thanks in advance and with kind regards
October 8th, 2010, 03:41 PM   #2
Math Team

Joined: Dec 2006
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Re: rectangulars fit in one rectangular

Hello, PGO!

Quote:
 $\text{I'm trying to find to a formula for the following problem.}$ $\text{I have one big rectangle with fixed dimensions: }\:(a\,\times\,b)$ $\text{I have 10 other rectangles with variable dimensions: }\:(x_1,\,y_1),\;(x_2,\,y_2),\;(x_3,\,y_3),\;.\;.\ ;.\;(x_{10},\,y_{10})$ $\text{What is the formula to see if the 10 rectangles will fit in the big one?}$ [color=beige]. . [/color]$\text{Note: all 10 rectangles may be turned and combined with the others.}$

This is an extremely complex problem . . . with no simple formula.

It depends on the dimensions of the the big rectangle
[color=beige]. . [/color]and the dimensions of the ten small rectangles.
So we must consider twenty-two possible variables.
[color=beige]. . . [/color]Good luck!

Obviously, the combined areas of the ten small rectangles cannot exceed the area of the big rectangle.

[color=beige]. . [/color]$(x_1\,\times\,y_1)\,+\,(x_2\,\times\,y_2)\,+\,(x_3 \,\times\,y_3)\,+\:.\,.\,.\:+\,(x_{10}\,\times\,y_ {10}) \;\le\;(a\,\times\,b)$

Even then, we are not sure if the ten small rectangles will actually fit in the big one.

Suppose we had one each of the following ten rectangles:

[color=beige]. . [/color]$\begin{array}{c|c} \text{Dimensions} & \text{Area} \\\\ \hline \\ 1\times1 & 1 \\ \\ 1\times2 & 2 \\ \\ 1 \times3 & 3 \\ \\ 1\times4 & 4 \\ \\ 1\times5 & 5 \\ \\ 2 \times 2 & 4 \\ \\ 2\times 3 & 6 \\ \\ 2 \times 4 & 8 \\ \\ 2 \times 5 & 10 \\ \\ 3 \times 3 & 9 \\ \\ \hline \\ \text{Total:} & 52 \end{array}$

If the big rectangle is: $4\,\times\,13$, we might be able to fit in the 10 rectangles.

If the big rectangle is: $2\,\times\,26$, it is impossible.
(The $3\times 3$ won't fit, of course.)

 October 10th, 2010, 04:33 AM #3 Newbie   Joined: Oct 2010 Posts: 2 Thanks: 0 Re: rectangulars fit in one rectangular Hi, Thanks for your reply! Indeed the combined surfaces of the small rectangulars can't exceed the big one. However, we did not yet solve the problem of the other equations

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