To determine during which interval the weight is above its resting (equilibrium) position, we need to analyze the displacement values given in the table. The displacement values tell us the position of the weight at specific time instances. Specifically, we need to identify where the displacement values are positive, which indicates that the weight is above the resting position.
Let's look at the values given in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
t & 0 & 0.015625 & 0.03125 & 0.046875 & 0.0625 & 0.078125 & 0.09375 \\
\hline
d(t) & -20 & -14.14 & 0 & 14.14 & 20 & 14.14 & 0 \\
\hline
\end{array}
\][/tex]
We observe the following points:
- For [tex]\( t = 0 \)[/tex], [tex]\( d(t) = -20 \)[/tex] (below the resting position)
- For [tex]\( t = 0.015625 \)[/tex], [tex]\( d(t) = -14.14 \)[/tex] (below the resting position)
- For [tex]\( t = 0.03125 \)[/tex], [tex]\( d(t) = 0 \)[/tex] (at the resting position)
- For [tex]\( t = 0.046875 \)[/tex], [tex]\( d(t) = 14.14 \)[/tex] (above the resting position)
- For [tex]\( t = 0.0625 \)[/tex], [tex]\( d(t) = 20 \)[/tex] (above the resting position)
- For [tex]\( t = 0.078125 \)[/tex], [tex]\( d(t) = 14.14 \)[/tex] (above the resting position)
- For [tex]\( t = 0.09375 \)[/tex], [tex]\( d(t) = 0 \)[/tex] (at the resting position)
From this, it's clear that the weight is above its resting position (displacement is positive) when [tex]\( 0.046875 \le t < 0.078125 \)[/tex].
Now, let's match this interval with the given choices:
A. [tex]\((0, 0.03125)\)[/tex] â No, the weight is below or at the resting position during this interval.
B. [tex]\((0.03125, 0.0625)\)[/tex] â No, this interval includes where it starts to move above the resting position but isn't entirely above throughout.
C. [tex]\((0.046875, 0.078125)\)[/tex] â Yes, this matches our identified interval.
D. [tex]\((0.03125, 0.09375)\)[/tex] â No, this interval includes both positions below and above the resting position.
Therefore, the correct answer is:
C. [tex]\((0.046875, 0.078125)\)[/tex]
