A worker is asked to arrange 1000 identical square tiles into a rectangular pattern and paint only the tiles forming the border. What should be the dimension of the rectangular pattern he arranges, in order to use the minimum amount of paint?
Tile Arrangement for Minimum Paint
The problem asks us to arrange 1000 identical square tiles into a rectangular pattern such that the number of tiles forming the border is minimized. The worker needs to paint only these border tiles. Minimizing the amount of paint means minimizing the number of border tiles.
We are arranging 1000 tiles into a rectangle. This means the area of the rectangle is fixed at 1000 square units (since each tile is one square unit). Let the dimensions of the rectangle be \(l\) tiles by \(w\) tiles. The total number of tiles is given by the area:
\[ l \times w = 1000 \]
The tiles forming the border are the tiles on the outer edges of the rectangle. The number of border tiles can be calculated as the total number of tiles minus the number of tiles in the inner rectangle (the rectangle formed by removing the outer border). If the outer dimensions are \(l\) and \(w\), the inner dimensions are \((l-2)\) and \((w-2)\), assuming \(l \ge 2\) and \(w \ge 2\).
Number of border tiles = Total tiles - Inner tiles
Number of border tiles = \(lw - (l-2)(w-2)\)
Since \(lw = 1000\), this is \(1000 - (lw - 2l - 2w + 4) = 1000 - 1000 + 2l + 2w - 4 = 2l + 2w - 4\). This formula represents the perimeter \((2l + 2w)\) minus the 4 corner tiles which are counted twice in the perimeter sum.
To minimize the number of border tiles \((2l + 2w - 4)\), we need to minimize the perimeter \((2l + 2w)\), which is equivalent to minimizing the sum of the dimensions \((l+w)\), given that the product \(lw\) is constant (1000).
For a fixed area, the sum of the dimensions \((l+w)\) is minimized when the rectangle is as close to a square as possible. We need to find pairs of factors \((l, w)\) of 1000 that are closest to each other. The square root of 1000 is approximately 31.6.
Let's look at the given options for the dimensions and calculate the sum of the dimensions and the number of border tiles for each option.
| Option | Dimensions (\(l \times w\)) | Sum of Dimensions (\(l+w\)) | Number of Border Tiles (\(2l + 2w - 4\)) |
|---|---|---|---|
| 1 | 50 tiles \(\times\) 20 tiles | \(50 + 20 = 70\) | \(2(50) + 2(20) - 4 = 100 + 40 - 4 = 136\) |
| 2 | 8 tiles \(\times\) 125 tiles | \(8 + 125 = 133\) | \(2(8) + 2(125) - 4 = 16 + 250 - 4 = 262\) |
| 3 | 200 tiles \(\times\) 5 tiles | \(200 + 5 = 205\) | \(2(200) + 2(5) - 4 = 400 + 10 - 4 = 406\) |
| 4 | 40 tiles \(\times\) 25 tiles | \(40 + 25 = 65\) | \(2(40) + 2(25) - 4 = 80 + 50 - 4 = 126\) |
Comparing the sums of dimensions, the minimum sum is 65, corresponding to the dimensions 40 tiles \(\times\) 25 tiles. This also results in the minimum number of border tiles, which is 126.
Thus, the dimensions that require the minimum amount of paint for the border are 40 tiles \(\times\) 25 tiles.
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