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 $x$-value, the rockets followed the same path.
B. Since the $x$-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.
Answer (1)
Both rockets launch and land at the same times, indicating equal time in the air.
Rocket B reaches a higher maximum height (100 ft) compared to Rocket A (75 ft).
The x -intercepts being the same does not imply the same path due to differing heights.
Therefore, the rocket for group B went higher in the air, while both rockets were in the air for the same amount of time. The rockets were in the air for the same amount of time, but the rocket for group B went higher in the air.
Explanation
Understanding the Problem We are given data for the height of two rockets over time. We need to compare their paths based on the given options.
Analyzing Lab Group A's Data Let's analyze the data for Lab Group A. The rocket starts at 0 ft at 0 seconds, reaches a maximum height of 75 ft at 5 seconds, and returns to 0 ft at 10 seconds.
Analyzing Lab Group B's Data Now let's analyze the data for Lab Group B. The rocket starts at 0 ft at 0 seconds, reaches a maximum height of 100 ft at 5 seconds, and returns to 0 ft at 10 seconds.
Comparing the Data Comparing the two sets of data, we can see that both rockets were in the air for the same amount of time (10 seconds). However, the rocket for Lab Group B reached a higher maximum height (100 ft) than the rocket for Lab Group A (75 ft).
Evaluating the Options Now let's evaluate the given options:
"Since the maximums occurred at the same x -value, the rockets followed the same path." This is incorrect because the maximum heights are different.
"Since the x -intercepts are the same for both functions, the rockets followed the same path." This is incorrect because the maximum heights are different.
"The rockets were in the air for the same amount of time, but the rocket for group B went higher in the air." This is correct.
"The rockets traveled the same height in the air, but the rocket for group B was in the air for a longer time." This is incorrect because the rocket for group B was not in the air for a longer time, and they did not travel the same height.
Final Answer Therefore, the correct answer is: The rockets were in the air for the same amount of time, but the rocket for group B went higher in the air.
Examples
Understanding the paths of rockets can be applied to various real-world scenarios, such as designing trajectories for satellites or analyzing the flight paths of projectiles. By comparing the height and time data of different rockets, scientists and engineers can optimize rocket designs and improve the accuracy of space missions. This type of analysis is also crucial in fields like aerospace engineering and ballistics, where precise control over projectile motion is essential. For example, if you are designing a firework display, you need to know how high each firework will go and how long it will be in the air to create a visually appealing show.
