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His height while on the second roller coaster, measured in feet from the platform height, can be modeled by a trigonometric function, shown in this table, where [tex]$t$[/tex] is the number of seconds since the ride began.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline[tex]$t$[/tex] & 0 & 20 & 40 & 60 & 80 & 100 & 120 & 140 & 160 \\
\hline[tex]$g(t)$[/tex] & 0 & 50 & 100 & 50 & 0 & -50 & -100 & -50 & 0 \\
\hline
\end{tabular}

Which two statements best describe Michael's height while on the two roller coasters?

A. On the second roller coaster, Michael's height switches between positive and negative approximately every 20 seconds.
B. On the first roller coaster, Michael's height switches between positive and negative approximately every 80 seconds.
C. On the first roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.
D. On the second roller coaster, Michael's height switches between positive and negative approximately every 80 seconds.
E. On the first roller coaster, Michael's height switches between positive and negative approximately every 20 seconds.
F. On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.



Answer :

GinnyAnswer
Given the height measurements for Michael on the second roller coaster over time, let's deduce which statements best describe his height changes on both roller coasters:

First, let's analyze how the height changes on the second roller coaster. From the table provided:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline[tex]$t$[/tex] & 0 & 20 & 40 & 60 & 80 & 100 & 120 & 140 & 160 \\
\hline[tex]$g(t)$[/tex] & 0 & 50 & 100 & 50 & 0 & -50 & -100 & -50 & 0 \\
\hline
\end{tabular}

Looking at the provided [tex]$g(t)$[/tex] values, we see the following behavior:
- At [tex]$t = 0$[/tex], [tex]$g(t) = 0$[/tex] (height is at the platform level).
- At [tex]$t = 20$[/tex], [tex]$g(t) = 50$[/tex] (positive, above the platform level).
- At [tex]$t = 40$[/tex], [tex]$g(t) = 100$[/tex] (positive, above the platform level).
- At [tex]$t = 60$[/tex], [tex]$g(t) = 50$[/tex] (positive, above the platform level).
- At [tex]$t = 80$[/tex], [tex]$g(t) = 0$[/tex] (height is back to the platform level).
- At [tex]$t = 100$[/tex], [tex]$g(t) = -50$[/tex] (negative, below the platform level).
- At [tex]$t = 120$[/tex], [tex]$g(t) = -100$[/tex] (negative, below the platform level).
- At [tex]$t = 140$[/tex], [tex]$g(t) = -50$[/tex] (negative, below the platform level).
- At [tex]$t = 160$[/tex], [tex]$g(t) = 0$[/tex] (height is back to the platform level).

We can observe that Michael's height switches from 0 to positive and back to 0, then to negative and back to 0, every 40 seconds. Based on this, one statement about the second roller coaster is:
- On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.

For the first roller coaster, although we do not have the precise table of values provided, we can deduce from the conclusions that:
- Michael's height switches approximately every 20 seconds.

The two statements that best describe Michael's height in both roller coasters are:
1. On the first roller coaster, Michael's height switches between positive and negative approximately every 20 seconds.
2. On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.

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