To determine which table will fit snugly in the corner of a room where the walls meet at a [tex]\(90^\circ\)[/tex] angle, we need to calculate or consider the diagonal length of each table and compare these measurements.
Given data for the tables:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
\text{Table} & \text{Length (feet)} & \text{Width (feet)} & \text{Diagonal (feet)} \\
\hline
1 & 1.5 & 2.4 & 3.0 \\
2 & 2.4 & 3.2 & 4.0 \\
3 & 3.2 & 2.8 & 4.0 \\
4 & 2.6 & \text{unknown} & \text{unknown} \\
\hline
\end{array}
\][/tex]
To fit snugly in a corner, the table must have the smallest diagonal length. The table measurements already given include calculated diagonals for Table 1, Table 2, and Table 3. For Table 4, we need to calculate the diagonal.
The formula for the diagonal [tex]\(d\)[/tex] of a rectangle when the length ([tex]\(l\)[/tex]) and width ([tex]\(w\)[/tex]) are known is:
[tex]\[
d = \sqrt{l^2 + w^2}
\][/tex]
For Table 4:
- Length [tex]\(l = 2.6\)[/tex] feet
- Width [tex]\(w\)[/tex] is unknown, but we cannot calculate the diagonal without [tex]\(w\)[/tex]. Thus, we must disregard Table 4 for now.
The known diagonal lengths are:
- Table 1: 3.0 feet
- Table 2: 4.0 feet
- Table 3: 4.0 feet
Since Table 4 cannot be considered due to missing data and among the other tables, Table 1 has the smallest diagonal length (3.0 feet).
Therefore, the table that fits most snugly in the corner is:
[tex]\[
\boxed{\text{Table 1}}
\][/tex]
