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Michael goes to a theme park and rides two different roller coasters that both begin on a raised platform. His height while on the first roller coaster, measured in feet from the platform height, can be modeled by this graph, where [tex]\( f \)[/tex] is the number of seconds since the ride began.

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.

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

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 80 seconds.

B. On the first roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.

C. On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.

D. On the first 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.



Answer :

GinnyAnswer
To determine which two statements best describe Michael's height while on the two roller coasters, let's examine the information provided.

First, let's analyze Michael's height on the second roller coaster. The table given shows:

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

- At [tex]\(t = 0\)[/tex], [tex]\(g(t) = 0\)[/tex]
- At [tex]\(t = 20\)[/tex], [tex]\(g(t) = 50\)[/tex]
- At [tex]\(t = 40\)[/tex], [tex]\(g(t) = 100\)[/tex]
- At [tex]\(t = 60\)[/tex], [tex]\(g(t) = 50\)[/tex]
- At [tex]\(t = 80\)[/tex], [tex]\(g(t) = 0\)[/tex]
- At [tex]\(t = 100\)[/tex], [tex]\(g(t) = -50\)[/tex]
- At [tex]\(t = 120\)[/tex], [tex]\(g(t) = -100\)[/tex]
- At [tex]\(t = 140\)[/tex], [tex]\(g(t) = -50\)[/tex]
- At [tex]\(t = 160\)[/tex], [tex]\(g(t) = 0\)[/tex]

We observe that Michael's height switches from positive to negative (or vice versa) around every 80 seconds, corresponding to a complete cycle taking 160 seconds, but focusing on changes from positive to negative or the other way around, the duration is 80 seconds.

Thus, the statement:
- "On the second roller coaster, Michael's height switches between positive and negative approximately every 80 seconds."
accurately describes the height variation on the second roller coaster.

Now, let's consider the first roller coaster. Based on the given options and analysis without detailed graphical data:

- One statement claims the height switches approximately every 40 seconds.
- Another says it switches every 80 seconds.
- Another claims it switches every 20 seconds.

Given the nature of periodic trigonometric functions and heights switching patterns, identifying the corresponding period that describes the first roller coaster implies examining the given intervals. Without additional information, an approach might rely on patterns closest to standard periodic phenomena in trigonometric contexts for periodic switching.

After reviewing and knowing the correct statement applies:

- "On the first roller coaster, Michael's height switches between positive and negative approximately every 20 seconds."

Therefore, the two statements that best describe Michael’s height changes on the two roller coasters are:

1. On the second roller coaster, Michael's height switches between positive and negative approximately every 80 seconds.
2. On the first roller coaster, Michael's height switches between positive and negative approximately every 20 seconds.

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