Let's break this problem down step-by-step based on the provided table and the required results.
1. Identifying the time interval Riley is moving forward:
From the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
t & 0 & 0.75 & 1.5 & 2.25 & 3 & 3.75 & 4.5 & 5.25 & 6 & 6.75 \\
\hline
f(t) & 0 & 38.9 & 55 & 38.9 & 0 & -28.9 & -55 & -38.9 & 0 & 38.9 \\
\hline
\end{array}
\][/tex]
Riley moves forward from the starting point ([tex]$f(t) = 0$[/tex]) to the maximum forward swing when [tex]$f(t)$[/tex] starts to be positive until it becomes zero or negative. Observing the values in the [tex]\(f(t)\)[/tex] row, Riley first reaches a non-positive value at [tex]$t = 3$[/tex] seconds. Therefore, the interval where Riley is moving forward is from [tex]\([0, 2.25]\)[/tex] seconds.
2. Calculating the total swing cycle time:
Riley starts swinging from [tex]$t = 0$[/tex] seconds, goes forward, reaches the furthest backward position and returns back to the starting position at [tex]\(t = 6.75\)[/tex]. Thus, it takes Riley [tex]\(6.75\)[/tex] seconds to complete one full cycle of swinging forward, back, and returning to the starting position.
3. Finding the maximum distance:
Reviewing the [tex]\(f(t)\)[/tex] values, the maximum positive value occurs at [tex]$t = 1.5$[/tex] seconds, where [tex]\(f(1.5) = 55\)[/tex] inches. This is the furthest distance Riley swings from her starting position in the positive direction.
Therefore, the selections for the drop-down menus should be:
- Riley is moving forward on the interval [tex]\([0, 2.25]\)[/tex] seconds.
- It takes Riley [tex]\(6.75\)[/tex] seconds to swing forward, back, and then return to her starting position.
- Riley reaches a maximum distance of [tex]\(55\)[/tex] inches from her starting position.
Thus, the completed statements will be:
- From the table, Riley is moving forward on the interval [tex]\([0, 2.25]\)[/tex].
- It takes Riley [tex]\(6.75\)[/tex] seconds to swing forward, back, and then return to her starting position.
- Riley reaches a maximum distance of [tex]\(55\)[/tex] inches from her starting position.
