The following tile floor pattern is made of 46 square tiles The tires form a rectangle the length of the long side of the rectangle is 42.75 in what is the side length in inches of one tile

To find the side length of one tile, we first need to determine the dimensions of the rectangle formed by the tiles. We know that the area of the rectangle (in terms of the number of tiles) is given as 46 square tiles, and that the longer side of the rectangle has a length of 42.75 inches.

Let's denote the side length of one tile as \( s \). The area of the rectangle in terms of \( s \) is:

\[
\text{Area} = \text{length} \times \text{width} = 42.75 \times w \quad (\text{where } w \text{ is the width})
\]

Since there are 46 tiles in total, the total area can also be expressed as:

\[
\text{Area} = 46 \times s^2
\]

So we can set the two expressions for area equal to each other:

\[
42.75 \times w = 46 \times s^2
\]

Next, we need to determine \( w \). The width can be expressed in terms of the length of one tile:

\[
w = \frac{46 \times s^2}{42.75}
\]

Now, since the area must be equal to the area formed by the whole number of tiles, also note that \( w \) must be an integer. Since the length of the rectangle (42.75 inches) is fixed, we can deduce that the width must neatly fit tiles across.

Assuming the tiles are square, the rectangle can be arranged in terms of both \( s \) (the side length of one tile) and the number of tiles along the width. Let’s say the number of tiles along the width is \( n \). The relationship now becomes:

\[
42.75 = n \cdot s
\]

Also, we can express the total number of tiles in terms of the dimensions \( n \) (width) and \( m \) (length):

\[
n \cdot m = 46
\]

Assuming \( m \) (the number of tiles along the longer side) corresponds to the length of 42.75, we can write:

1. \( n \cdot m = 46 \)
2. \( m = \frac{42.75}{s} \)

Substituting (2) into (1):

\[
n \cdot \left(\frac{42.75}{s}\right) = 46
\]

This simplifies to:

\[
n \cdot 42.75 = 46s
\]

Thus we have:

\[
s = \frac{n \cdot 42.75}{46}
\]

Since \( n \) must be a divisor of 46 for \( m \) to be an integer, the divisors of 46 are 1, 2, 23, and 46. We can try these values of \( n \).

1. **If \( n = 1 \)**:
\[
s = \frac{1 \cdot 42.75}{46} = \frac{42.75}{46} \approx 0.9283
\]

2. **If \( n = 2 \)**:
\[
s = \frac{2 \cdot 42.75}{46} = \frac{85.5}{46} \approx 1.8577
\]

3. **If \( n = 23 \)**:
\[
s = \frac{23 \cdot 42.75}{46} = \frac{983.25}{46} \approx 21.4011
\]

4. **If \( n = 46 \)**:
\[
s = \frac{46 \cdot 42.75}{46} = 42.75
\]

Only the values of \( s \) that make sense in context are \( \approx 0.9283 \) and \( \approx 1.8577 \), as the others do not match realistically as tile sizes.

To further resolve which of these values corresponds to a tile size that fits within the distribution of the tiles in the layout, if \( n = 2\) (and thus \( m = 23\)), we obtain:

Thus, we could infer:

The side length of one tile is approximately:

\[
\boxed{1.85 \text{ inches}}
\]