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The tables show functions representing the path of two rockets launched by different science lab groups.

Lab Group A
\begin{tabular}{|l|c|c|c|c|c|}
\hline Time after Launch (sec) & 0 & 2 & 5 & 8 & 10 \\
\hline Height (ft) & 0 & 48 & 75 & 48 & 0 \\
\hline
\end{tabular}

Lab Group B
\begin{tabular}{|l|c|c|c|c|c|}
\hline Time after Launch (sec) & 0 & 1 & 5 & 9 & 10 \\
\hline Height (ft) & 0 & 36 & 100 & 36 & 0 \\
\hline
\end{tabular}

How do the functions in the tables compare?

A. Since the maximums occurred at the same [tex]$x$[/tex]-value, the rockets followed the same path.
B. Since the [tex]$x$[/tex]-intercepts are the same for both functions, the rockets followed the same path.
C. The rockets were in the air for the same amount of time, but the rocket for group B went higher in the air.
D. The rockets traveled the same height in the air, but the rocket for group B was in the air for a longer time.

Sagot :

GinnyAnswer
To compare the functions that represent the paths of the two rockets launched by different lab groups, we need to go through a step-by-step analysis comparing their heights at different times, the maximum heights achieved, and the times the rockets were in the air.

Let's start by understanding the given data:

### Lab Group A Rocket Path
| Time after Launch (sec) | 0 | 2 | 5 | 8 | 10 |
|--------------------------|---|---|---|---|----|
| Height (ft) | 0 | 48 | 75 | 48 | 0 |

### Lab Group B Rocket Path
| Time after Launch (sec) | 0 | 1 | 5 | 9 | 10 |
|--------------------------|---|---|---|---|----|
| Height (ft) | 0 | 36 | 100 | 36 | 0 |

#### Analysis:

1. Initial and Final Heights (x-intercepts):
- Both rockets start at height 0 ft and end at height 0 ft.
- Time of launch and landing for both rockets is from second 0 to second 10.

2. Maximum Heights:
- Lab Group A: Maximum height is 75 ft at time 5 seconds.
- Lab Group B: Maximum height is 100 ft at time 5 seconds.

3. Times of Maximum Heights:
- Both rockets reach their maximum heights at time 5 seconds.

4. Total Time in the Air:
- Both rockets are in the air for a total of 10 seconds (from launch to landing at t = 0 to t = 10).

Based on this analysis, we can conclude:

- Since the maximum heights occurred at the same time (x-value), it means the maximum heights were attained at the same time.
- The x-intercepts are the same for both functions, indicating that both rockets were launched and landed at the same times.

Thus, the conclusion based on the given analysis is:

Since the [tex]$x$[/tex]-intercepts are the same for both functions, the rockets followed the same path.

Therefore, the correct answer is:

```plaintext
Since the [tex]$x$[/tex]-intercepts are the same for both functions, the rockets followed the same path.
```

This matches option 2 in the provided choices.
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