To determine how long it takes for the friend in the left seat to move upward, downward, and then back to the initial position, we need to analyze the given data in the table showing the height of the left seat over time.
The height, [tex]\( h(t) \)[/tex], at different times [tex]\( t \)[/tex] is provided as follows:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
t & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
h(t) & 24 & 48 & 24 & 0 & 24 & 48 & 24 \\
\hline
\end{array}
\][/tex]
From this data, we can observe the behavior of the height over time:
- At [tex]\( t = 0 \)[/tex] seconds, the height [tex]\( h(0) = 24 \)[/tex] inches.
- At [tex]\( t = 1 \)[/tex] second, the height [tex]\( h(1) = 48 \)[/tex] inches.
- At [tex]\( t = 2 \)[/tex] seconds, the height [tex]\( h(2) = 24 \)[/tex] inches.
- At [tex]\( t = 3 \)[/tex] seconds, the height [tex]\( h(3) = 0 \)[/tex] inches.
- At [tex]\( t = 4 \)[/tex] seconds, the height [tex]\( h(4) = 24 \)[/tex] inches.
- At [tex]\( t = 5 \)[/tex] seconds, the height [tex]\( h(5) = 48 \)[/tex] inches.
- At [tex]\( t = 6 \)[/tex] seconds, the height [tex]\( h(6) = 24 \)[/tex] inches.
We see that the pattern repeats every 6 seconds. At [tex]\( t = 6 \)[/tex] seconds, the seat height returns to the initial height of 24 inches, completing one full seesaw cycle (upward and downward motion back to the initial position).
Therefore, the time taken for the friend in the left seat to move upward, downward, and back to the initial position is [tex]\( \boxed{6} \)[/tex] seconds.
