Let's consider the equation:
[tex]\[
\sqrt{2 x + 3} = \frac{\pi}{x + 5} + 2
\][/tex]
We need to find the row in the table where the values of [tex]\( \sqrt{2 x + 3} \)[/tex] and [tex]\( \frac{\pi}{x + 5} + 2 \)[/tex] are closest to each other.
Given the table:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
$x$ & $\sqrt{2 x+3}$ & $\frac{\pi}{x+5}+2$ \\
\hline
0 & 1.7321 & 2 \\
\hline
0.1 & 1.7889 & 2.0196 \\
\hline
0.2 & 1.8439 & 2.0385 \\
\hline
0.3 & 1.8974 & 2.0566 \\
\hline
0.4 & 1.9494 & 2.0741 \\
\hline
0.5 & 2 & 2.0909 \\
\hline
0.6 & 2.0494 & 2.1071 \\
\hline
0.7 & 2.0976 & 2.1228 \\
\hline
0.8 & 2.1448 & 2.1379 \\
\hline
0.9 & 2.1909 & 2.1525 \\
\hline
1.0 & 2.2361 & 2.5236 \\
\hline
\end{tabular}
\][/tex]
We need to determine which row has the values of [tex]\( \sqrt{2 x + 3} \)[/tex] and [tex]\( \frac{\pi}{x + 5} + 2 \)[/tex] closest to each other.
After examining the table, we find:
- For [tex]\( x = 1.0 \)[/tex]:
[tex]\[
\sqrt{2 \times 1.0 + 3} = 2.2361
\][/tex]
[tex]\[
\frac{\pi}{1.0 + 5} + 2 \approx 2.5236
\][/tex]
These values are the closest to each other compared to the other rows.
Thus, the row in the table that is closest to the actual solution of the equation is:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
1.0 & 2.2361 & 2.5236 \\
\hline
\end{tabular}
\][/tex]
This corresponds to the row with [tex]\( x = 1.0 \)[/tex].
