Question
The table relates to a function h(t) that models the height of a ball t seconds after it is dropped.
A 2-column table with 6 rows. The first column is labeled x with entries 0, 1, 2, 3, 4, 5, 6. The second column is labeled h(t) with entries 150, 145.1, 130.4, 105.9, 71.6, 27.5, negative 26.4.
Determine the initial height of the ball and the time interval before the ball hits the ground.
initial height = 0; hits the ground between 5 and 6 seconds
initial height = 150; hits the ground between 5 and 6 seconds
initial height = 0; hits the ground between 3 and 4 seconds
initial height = 150; hits the ground between 4 and 5 second
A 2-column table with 6 rows. The first column is labeled x with entries 0, 1, 2, 3, 4, 5, 6. The second column is labeled h(t) with entries 150, 145.1, 130.4, 105.9, 71.6, 27.5, negative 26.4.
Determine the initial height of the ball and the time interval before the ball hits the ground.
initial height = 0; hits the ground between 5 and 6 seconds
initial height = 150; hits the ground between 5 and 6 seconds
initial height = 0; hits the ground between 3 and 4 seconds
initial height = 150; hits the ground between 4 and 5 second
Answers
GPT-4o
To determine the initial height of the ball and the time interval before the ball hits the ground, we need to analyze the given table:
\[
\begin{array}{c|c}
t & h(t) \\
\hline
0 & 150 \\
1 & 145.1 \\
2 & 130.4 \\
3 & 105.9 \\
4 & 71.6 \\
5 & 27.5 \\
6 & -26.4 \\
\end{array}
\]
1. **Initial Height**: The initial height of the ball is the height at \( t = 0 \). Looking at the table, when \( t = 0 \), \( h(t) = 150 \). Thus, the initial height is 150 units.
2. **Time Interval When the Ball Hits the Ground**: The ball hits the ground when its height \( h(t) \) reaches 0. From the table:
- At \( t = 5 \), \( h(t) = 27.5 \).
- At \( t = 6 \), \( h(t) = -26.4 \).
The height changes from 27.5 at 5 seconds to -26.4 at 6 seconds, meaning the ball must have hit the ground (where height \( h(t) = 0 \)) sometime between 5 and 6 seconds.
Therefore, the correct answer is:
**initial height = 150; hits the ground between 5 and 6 seconds**.
\[
\begin{array}{c|c}
t & h(t) \\
\hline
0 & 150 \\
1 & 145.1 \\
2 & 130.4 \\
3 & 105.9 \\
4 & 71.6 \\
5 & 27.5 \\
6 & -26.4 \\
\end{array}
\]
1. **Initial Height**: The initial height of the ball is the height at \( t = 0 \). Looking at the table, when \( t = 0 \), \( h(t) = 150 \). Thus, the initial height is 150 units.
2. **Time Interval When the Ball Hits the Ground**: The ball hits the ground when its height \( h(t) \) reaches 0. From the table:
- At \( t = 5 \), \( h(t) = 27.5 \).
- At \( t = 6 \), \( h(t) = -26.4 \).
The height changes from 27.5 at 5 seconds to -26.4 at 6 seconds, meaning the ball must have hit the ground (where height \( h(t) = 0 \)) sometime between 5 and 6 seconds.
Therefore, the correct answer is:
**initial height = 150; hits the ground between 5 and 6 seconds**.