To find the side length of one tile, we can start by determining the dimensions of the rectangle formed by the 36 tiles.
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Calculate the area of the rectangle: The area is given by the product of the length and the width. We know the length of the longer side is 42.75 inches, and we need to find the width using the total area of the tiles.
The total area of the rectangle is the area per tile multiplied by the number of tiles:
\[ \text{Total Area} = 36 \cdot (\text{side length of one tile})^2 \]
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Find the total area from the longer side's length: Let \( w \) represent the width of the rectangle. The area of the rectangle can also be expressed as:
\[ \text{Area} = \text{length} \times \text{width} = 42.75 \times w \]
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Set the two area expressions equal to each other:
\[ 36 \cdot (\text{side length of one tile})^2 = 42.75 \times w \]
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Determine possible dimensions for the rectangle: Since there are 36 tiles, and they must be arranged in a rectangular shape, we need to find two factors of 36. The possible factor pairs (length, width) could be:
- \( (1, 36) \)
- \( (2, 18) \)
- \( (3, 12) \)
- \( (4, 9) \)
- \( (6, 6) \)
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Assume \( w \) is one of these factor pairs: Let's first calculate the possible widths from factor pairs:
- If \( w = 1 \) -> Length = 36
- If \( w = 2 \) -> Length = 18
- If \( w = 3 \) -> Length = 12
- If \( w = 4 \) -> Length = 9
- If \( w = 6 \) -> Length = 6
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Determine side length when \( w \) is substituted into the area equation: Each of these combinations will yield different side lengths. We'll substitute \( w \) back into the rectangle's area equation to find the side length of each tile.
Since the longer length is given as 42.75 inches, we won't directly calculate the pairs but will consider inferences about the short side:
- Assume \( w \) is among these pairs and is the short side. Because we do not get perfect pairs, we can also estimate the side of the tile directly.
Assume square tile sides. The actual total tile count does not perfectly equal a direct multiplication for getting tile widths.
Thus we calculate using:
Let's calculate for \( 4 \) and \( 9 = 36 \)
Trying direct calculations, we note that: \[ \text{long side} (42.75) = 4 \times (n) \] for the short side being 9 roots to give:
\[ \text{length} = \text{side} \times 9 \]
By iterating,
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Each factor from \(36\), Let assume while dividing:
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\( \text{side length} = 4.75\) will be computed out side \( 9 \times (4.75\)
Which yields combinations like \( 36\), or tile area \( = 42.75\).
Lastly reviewing choices we see with tile: The aforementioned dimension which are multiples... all indicate the correct option leads towards:
\[ \text{Tile length} = 4.75 \]
Thus, the side length of one tile overall would be: The correct answer is 4.75 inches.