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Antoine pulls a weight attached to a spring below its resting, or equilibrium, position. When Antoine releases the weight, it oscillates above and below its equilibrium position as shown in the table. The displacement of the weight from its equilibrium position, [tex]\( d(t) \)[/tex], in centimeters, is modeled by a cosine function, where [tex]\( t \)[/tex] represents time, in seconds.

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
t & 0 & 0.015625 & 0.03125 & 0.046875 & 0.0625 & 0.078125 & 0.09375 \\
\hline
d(t) & -20 & -14.14 & 0 & 14.14 & 20 & 14.14 & 0 \\
\hline
\end{tabular}
\][/tex]

At which interval is the weight above its resting position?

A. [tex]\((0.046875, 0.078125)\)[/tex]

B. [tex]\((0.09125, 0.0025)\)[/tex]

C. [tex]\((0, 0.03125)\)[/tex]

D. [tex]\((0.03125, 0.09975)\)[/tex]



Answer :

GinnyAnswer
To determine the interval during which the weight is above its resting position, we need to analyze the provided data for [tex]\( d(t) \)[/tex], the displacement in centimeters, at specific times [tex]\( t \)[/tex], measured in seconds.

The given table is:

[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 want to find when [tex]\( d(t) > 0 \)[/tex]. Observing the values of [tex]\( d(t) \)[/tex] in the table:

1. [tex]\( t = 0: d(0) = -20 \)[/tex] (below equilibrium)
2. [tex]\( t = 0.015625: d(0.015625) = -14.14 \)[/tex] (below equilibrium)
3. [tex]\( t = 0.03125: d(0.03125) = 0 \)[/tex] (at equilibrium)
4. [tex]\( t = 0.046875: d(0.046875) = 14.14 \)[/tex] (above equilibrium)
5. [tex]\( t = 0.0625: d(0.0625) = 20 \)[/tex] (above equilibrium)
6. [tex]\( t = 0.078125: d(0.078125) = 14.14 \)[/tex] (above equilibrium)
7. [tex]\( t = 0.09375: d(0.09375) = 0 \)[/tex] (at equilibrium)

The weight is above its resting position (i.e., [tex]\( d(t) > 0 \)[/tex]) between the intervals of [tex]\( t \)[/tex]:

- Between [tex]\( 0.03125 \)[/tex] and [tex]\( 0.09375 \)[/tex].

From the options given, the correct interval would be:

D. [tex]\((0.03125, 0.09375)\)[/tex]

Thus, the correct answer is:
D. [tex]\((0.03125, 0.09375)\)[/tex]

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