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To determine which table most likely models the height [tex]\( h(t) \)[/tex] of a ball dropped from above ground level that hits the ground sometime between 4 and 6 seconds after it is dropped, we need to carefully examine each table and look for the table where the height [tex]\( h(t) \)[/tex] falls to zero (or below) within the specified time interval of 4 to 6 seconds.
Here are the given tables again for reference:
Table 1:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & 70 \\ \hline 2 & 50.4 \\ \hline 4 & -8.4 \\ \hline 6 & -106.4 \\ \hline \end{array} \][/tex]
Table 2:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & 100 \\ \hline 2 & 80.4 \\ \hline 4 & 21.6 \\ \hline 6 & -76.4 \\ \hline \end{array} \][/tex]
Table 3:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & -150 \\ \hline 2 & -130.4 \\ \hline 4 & -71.6 \\ \hline 6 & 26.4 \\ \hline \end{array} \][/tex]
Table 4:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & 180 \\ \hline 2 & 160.4 \\ \hline 4 & 101.6 \\ \hline 6 & 3.6 \\ \hline \end{array} \][/tex]
Now, let's evaluate each table individually to see which one meets the condition of the ball hitting the ground (i.e., height [tex]\( h(t) \)[/tex] being less than or equal to zero) between 4 and 6 seconds:
- Table 1:
- At [tex]\( t = 0 \)[/tex], [tex]\( h(t) = 70 \)[/tex] meters.
- At [tex]\( t = 2 \)[/tex], [tex]\( h(t) = 50.4 \)[/tex] meters.
- At [tex]\( t = 4 \)[/tex], [tex]\( h(t) = -8.4 \)[/tex] meters (less than zero, implying the ball hits the ground).
- At [tex]\( t = 6 \)[/tex], [tex]\( h(t) = -106.4 \)[/tex] meters (also less than zero, but the ball has already hit the ground by [tex]\( t = 4 \)[/tex]).
- Table 2:
- At [tex]\( t = 0 \)[/tex], [tex]\( h(t) = 100 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( h(t) = 80.4 \)[/tex]
- At [tex]\( t = 4 \)[/tex], [tex]\( h(t) = 21.6 \)[/tex]
- At [tex]\( t = 6 \)[/tex], [tex]\( h(t) = -76.4 \)[/tex] (less than zero, implying the ball hits the ground after [tex]\( t = 4 \)[/tex])
- Table 3:
- At [tex]\( t = 0 \)[/tex], [tex]\( h(t) = -150 \)[/tex] meters (already below zero, implying it starts below ground, which doesn't make sense for a dropped ball scenario).
- The rest of the heights at [tex]\( t = 2, 4, \)[/tex] and [tex]\( 6 \)[/tex] remain below zero, indicating incorrect initial conditions.
- Table 4:
- At [tex]\( t = 0 \)[/tex], [tex]\( h(t) = 180 \)[/tex] meters.
- At [tex]\( t = 2 \)[/tex], [tex]\( h(t) = 160.4 \)[/tex] meters.
- At [tex]\( t = 4 \)[/tex], [tex]\( h(t) = 101.6 \)[/tex] meters.
- At [tex]\( t = 6 \)[/tex], [tex]\( h(t) = 3.6 \)[/tex] meters (still above zero, not hitting the ground).
From the analysis, it is clear that Table 1 is the correct choice since it is the only table where the height [tex]\( h(t) \)[/tex] becomes less than zero (i.e., the ball hits the ground) between 4 and 6 seconds after being dropped. Thus, the ball hits the ground at [tex]\( t = 4 \)[/tex] seconds in Table 1. Therefore, the table that most likely relates to the function [tex]\( h(t) \)[/tex] is:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & 70 \\ \hline 2 & 50.4 \\ \hline 4 & -8.4 \\ \hline 6 & -106.4 \\ \hline \end{array} \][/tex]
So the answer is Table 1.
Here are the given tables again for reference:
Table 1:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & 70 \\ \hline 2 & 50.4 \\ \hline 4 & -8.4 \\ \hline 6 & -106.4 \\ \hline \end{array} \][/tex]
Table 2:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & 100 \\ \hline 2 & 80.4 \\ \hline 4 & 21.6 \\ \hline 6 & -76.4 \\ \hline \end{array} \][/tex]
Table 3:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & -150 \\ \hline 2 & -130.4 \\ \hline 4 & -71.6 \\ \hline 6 & 26.4 \\ \hline \end{array} \][/tex]
Table 4:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & 180 \\ \hline 2 & 160.4 \\ \hline 4 & 101.6 \\ \hline 6 & 3.6 \\ \hline \end{array} \][/tex]
Now, let's evaluate each table individually to see which one meets the condition of the ball hitting the ground (i.e., height [tex]\( h(t) \)[/tex] being less than or equal to zero) between 4 and 6 seconds:
- Table 1:
- At [tex]\( t = 0 \)[/tex], [tex]\( h(t) = 70 \)[/tex] meters.
- At [tex]\( t = 2 \)[/tex], [tex]\( h(t) = 50.4 \)[/tex] meters.
- At [tex]\( t = 4 \)[/tex], [tex]\( h(t) = -8.4 \)[/tex] meters (less than zero, implying the ball hits the ground).
- At [tex]\( t = 6 \)[/tex], [tex]\( h(t) = -106.4 \)[/tex] meters (also less than zero, but the ball has already hit the ground by [tex]\( t = 4 \)[/tex]).
- Table 2:
- At [tex]\( t = 0 \)[/tex], [tex]\( h(t) = 100 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( h(t) = 80.4 \)[/tex]
- At [tex]\( t = 4 \)[/tex], [tex]\( h(t) = 21.6 \)[/tex]
- At [tex]\( t = 6 \)[/tex], [tex]\( h(t) = -76.4 \)[/tex] (less than zero, implying the ball hits the ground after [tex]\( t = 4 \)[/tex])
- Table 3:
- At [tex]\( t = 0 \)[/tex], [tex]\( h(t) = -150 \)[/tex] meters (already below zero, implying it starts below ground, which doesn't make sense for a dropped ball scenario).
- The rest of the heights at [tex]\( t = 2, 4, \)[/tex] and [tex]\( 6 \)[/tex] remain below zero, indicating incorrect initial conditions.
- Table 4:
- At [tex]\( t = 0 \)[/tex], [tex]\( h(t) = 180 \)[/tex] meters.
- At [tex]\( t = 2 \)[/tex], [tex]\( h(t) = 160.4 \)[/tex] meters.
- At [tex]\( t = 4 \)[/tex], [tex]\( h(t) = 101.6 \)[/tex] meters.
- At [tex]\( t = 6 \)[/tex], [tex]\( h(t) = 3.6 \)[/tex] meters (still above zero, not hitting the ground).
From the analysis, it is clear that Table 1 is the correct choice since it is the only table where the height [tex]\( h(t) \)[/tex] becomes less than zero (i.e., the ball hits the ground) between 4 and 6 seconds after being dropped. Thus, the ball hits the ground at [tex]\( t = 4 \)[/tex] seconds in Table 1. Therefore, the table that most likely relates to the function [tex]\( h(t) \)[/tex] is:
[tex]\[ \begin{array}{|c|c|} \hline t & h(t) \\ \hline 0 & 70 \\ \hline 2 & 50.4 \\ \hline 4 & -8.4 \\ \hline 6 & -106.4 \\ \hline \end{array} \][/tex]
So the answer is Table 1.
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