Answer :
Based on the given table, we can analyze the height values, [tex]\(g(t)\)[/tex], at different times to determine the intervals at which Michael’s height switches between positive and negative on the second roller coaster.
Given data:
[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]
To understand the switching intervals, we should focus on the points where [tex]\(g(t)\)[/tex] crosses the x-axis, i.e., where [tex]\(g(t) = 0\)[/tex] or changes its sign.
Here's the step-by-step process to identify the intervals where the height switches between positive and negative:
1. Identify the points where [tex]\(g(t)\)[/tex] is zero:
[tex]\[ t = 0, 80, 160 \][/tex]
2. Identify the points where [tex]\(g(t)\)[/tex] switches from positive to negative or vice versa:
- From [tex]\(t = 0\)[/tex] to [tex]\(t = 20\)[/tex]: [tex]\(g(t)\)[/tex] changes from 0 to 50 (positive)
- From [tex]\(t = 20\)[/tex] to [tex]\(t = 40\)[/tex]: [tex]\(g(t)\)[/tex] changes from 50 to 100 (still positive)
- From [tex]\(t = 40\)[/tex] to [tex]\(t = 60\)[/tex]: [tex]\(g(t)\)[/tex] changes from 100 to 50 (still positive)
- From [tex]\(t = 60\)[/tex] to [tex]\(t = 80\)[/tex]: [tex]\(g(t)\)[/tex] changes from 50 to 0 (switches to zero here)
After [tex]\(t = 80\)[/tex], [tex]\(g(t)\)[/tex] then goes from negative to positive:
- From [tex]\(t = 80\)[/tex] to [tex]\(t = 100\)[/tex]: [tex]\(g(t)\)[/tex] changes from 0 to -50 (negative)
- From [tex]\(t = 100\)[/tex] to [tex]\(t = 120\)[/tex]: [tex]\(g(t)\)[/tex] changes from -50 to -100 (still negative)
- From [tex]\(t = 120\)[/tex] to [tex]\(t = 140\)[/tex]: [tex]\(g(t)\)[/tex] changes from -100 to -50 (still negative)
- From [tex]\(t = 140\)[/tex] to [tex]\(t = 160\)[/tex]: [tex]\(g(t)\)[/tex] changes from -50 to 0 (switches to zero here)
From the above observations, we see:
- The height switches between positive and negative every 40 seconds (from positive to negative at [tex]\(t = 40\)[/tex] seconds, and from negative to positive again at [tex]\(t = 80\)[/tex] seconds).
So, the second roller coaster's height switches between positive and negative every 40 seconds.
Therefore, given the statements:
- On the second roller coaster, Michael's height switches between positive and negative approximately every 20 seconds.
- On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.
The second statement is correct:
1. On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.
Now, examining the other roller coaster statements:
- Without data for the first roller coaster, we can't analyze its switching period. We're left with assumptions.
Given the data provided that the switch interval is 40 seconds for the second roller coaster, we inferatively choose not to speculate beyond given data.
So the best answers would be:
1. On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.
2. On the second roller coaster, Michael's height switches between positive and negative approximately every 20 seconds. (nearest given distractor)
But the second appears as redundant correctness hint - choose only statement:
- "On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds."
Would be the strictly undisputable sole conclusion.
Given data:
[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]
To understand the switching intervals, we should focus on the points where [tex]\(g(t)\)[/tex] crosses the x-axis, i.e., where [tex]\(g(t) = 0\)[/tex] or changes its sign.
Here's the step-by-step process to identify the intervals where the height switches between positive and negative:
1. Identify the points where [tex]\(g(t)\)[/tex] is zero:
[tex]\[ t = 0, 80, 160 \][/tex]
2. Identify the points where [tex]\(g(t)\)[/tex] switches from positive to negative or vice versa:
- From [tex]\(t = 0\)[/tex] to [tex]\(t = 20\)[/tex]: [tex]\(g(t)\)[/tex] changes from 0 to 50 (positive)
- From [tex]\(t = 20\)[/tex] to [tex]\(t = 40\)[/tex]: [tex]\(g(t)\)[/tex] changes from 50 to 100 (still positive)
- From [tex]\(t = 40\)[/tex] to [tex]\(t = 60\)[/tex]: [tex]\(g(t)\)[/tex] changes from 100 to 50 (still positive)
- From [tex]\(t = 60\)[/tex] to [tex]\(t = 80\)[/tex]: [tex]\(g(t)\)[/tex] changes from 50 to 0 (switches to zero here)
After [tex]\(t = 80\)[/tex], [tex]\(g(t)\)[/tex] then goes from negative to positive:
- From [tex]\(t = 80\)[/tex] to [tex]\(t = 100\)[/tex]: [tex]\(g(t)\)[/tex] changes from 0 to -50 (negative)
- From [tex]\(t = 100\)[/tex] to [tex]\(t = 120\)[/tex]: [tex]\(g(t)\)[/tex] changes from -50 to -100 (still negative)
- From [tex]\(t = 120\)[/tex] to [tex]\(t = 140\)[/tex]: [tex]\(g(t)\)[/tex] changes from -100 to -50 (still negative)
- From [tex]\(t = 140\)[/tex] to [tex]\(t = 160\)[/tex]: [tex]\(g(t)\)[/tex] changes from -50 to 0 (switches to zero here)
From the above observations, we see:
- The height switches between positive and negative every 40 seconds (from positive to negative at [tex]\(t = 40\)[/tex] seconds, and from negative to positive again at [tex]\(t = 80\)[/tex] seconds).
So, the second roller coaster's height switches between positive and negative every 40 seconds.
Therefore, given the statements:
- On the second roller coaster, Michael's height switches between positive and negative approximately every 20 seconds.
- On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.
The second statement is correct:
1. On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.
Now, examining the other roller coaster statements:
- Without data for the first roller coaster, we can't analyze its switching period. We're left with assumptions.
Given the data provided that the switch interval is 40 seconds for the second roller coaster, we inferatively choose not to speculate beyond given data.
So the best answers would be:
1. On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.
2. On the second roller coaster, Michael's height switches between positive and negative approximately every 20 seconds. (nearest given distractor)
But the second appears as redundant correctness hint - choose only statement:
- "On the second roller coaster, Michael's height switches between positive and negative approximately every 40 seconds."
Would be the strictly undisputable sole conclusion.
