Let's break down and analyze the given information to answer the question, "Who will have traveled for a greater amount of time when their distance from New York stops decreasing and starts increasing?"
Annabeth's Distance Model:
Annabeth's distance, [tex]\( D(t) \)[/tex], from New York is given by the function:
[tex]\[ D(t) = 60 \times |t - 3| \][/tex]
To understand when Annabeth's distance stops decreasing and starts increasing, we need to analyze the absolute value function. The critical point for the absolute value function [tex]\( |t - 3| \)[/tex] occurs at [tex]\( t = 3 \)[/tex]. This is where the function changes behavior.
Summary for Annabeth:
- Up to [tex]\( t = 3 \)[/tex]: Annabeth's distance decreases.
- After [tex]\( t = 3 \)[/tex]: Annabeth's distance increases.
- Therefore, Annabeth's distance from New York stops decreasing and starts increasing at [tex]\( t = 3 \)[/tex] hours.
Charlie's Distance Model:
Charlie's distance, [tex]\( F(t) \)[/tex], from New York is represented by the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
t & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline
F(t) & 270 & 180 & 90 & 0 & 90 & 180 \\
\hline
\end{array}
\][/tex]
From the table, we can observe how Charlie's distance changes over time:
- From [tex]\( t = 0 \)[/tex] to [tex]\( t = 6 \)[/tex]: Charlie's distance decreases from 270 miles to 0 miles.
- From [tex]\( t = 6 \)[/tex] to [tex]\( t = 10 \)[/tex]: Charlie's distance starts increasing again from 0 miles to 180 miles.
Summary for Charlie:
- Up to [tex]\( t = 6 \)[/tex]: Charlie's distance decreases.
- After [tex]\( t = 6 \)[/tex]: Charlie's distance increases.
- Therefore, Charlie's distance from New York stops decreasing and starts increasing at [tex]\( t = 6 \)[/tex] hours.
Comparison:
Now, we need to compare who traveled for a greater amount of time when their distance stopped decreasing and started increasing:
- Annabeth's critical time: [tex]\( t = 3 \)[/tex] hours
- Charlie's critical time: [tex]\( t = 6 \)[/tex] hours
Since [tex]\( 6 > 3 \)[/tex], Charlie has traveled for a greater amount of time when his distance from New York stops decreasing and starts increasing.
Conclusion:
The correct answer is:
A. Charlie
