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Sagot :
To determine which representation corresponds to a decreasing speed with an increasing time, let's break down the scenarios given for Simon and Raphael.
### Simon's Scenario
Simon drives faster as he enters the freeway from the entrance ramp. We are given the following data for Simon’s driving:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time} & \text{Speed} \\ \hline 0 & 0 \\ \hline 2 & 15 \\ \hline 4 & 25 \\ \hline 6 & 45 \\ \hline 8 & 70 \\ \hline \end{array} \][/tex]
To determine if Simon’s speed is decreasing with increasing time, we need to look at the changes in speed over time. We calculate the speed difference between consecutive times:
- From [tex]\( 0 \)[/tex] to [tex]\( 2 \)[/tex] seconds: [tex]\( 15 - 0 = 15 \)[/tex]
- From [tex]\( 2 \)[/tex] to [tex]\( 4 \)[/tex] seconds: [tex]\( 25 - 15 = 10 \)[/tex]
- From [tex]\( 4 \)[/tex] to [tex]\( 6 \)[/tex] seconds: [tex]\( 45 - 25 = 20 \)[/tex]
- From [tex]\( 6 \)[/tex] to [tex]\( 8 \)[/tex] seconds: [tex]\( 70 - 45 = 25 \)[/tex]
These differences indicate that Simon's speed is increasing over time because all the differences are positive.
To verify if the increments themselves are increasing, we review the differences between consecutive increments:
- From [tex]\( 15 \)[/tex] to [tex]\( 10 \)[/tex]: This decreases, which violates the condition for consistent increment.
- From [tex]\( 10 \)[/tex] to [tex]\( 20 \)[/tex]: This increases.
- From [tex]\( 20 \)[/tex] to [tex]\( 25 \)[/tex]: This increases.
Since there is an initial decrease in increments (from [tex]\( 15 \)[/tex] to [tex]\( 10 \)[/tex]), it's not a consistently increasing pattern of increments. But since all individual speed figures are increasing, Simon's speed indeed increases over time.
### Raphael's Scenario
Raphael rolls his ball downhill. When a ball rolls downhill without much or any control, typically its speed can vary depending on multiple factors like slope and friction. However, for the simplicity of this problem, we'll assume that "rolling downhill" implies that Raphael's speed is consistently decreasing with time as controlling speed may actually decrease due to various factors.
### Conclusion
Based on the provided prompt and analysis, here is the appropriate conclusion for each agent:
- For Simon's driving scenario, his speed increases over time rather than decreases.
- For Raphael's rolling ball scenario, it is reasonable to conclude that his speed decreases over time.
Thus, the correct representation corresponding to a decreasing speed with increasing time is for Raphael.
### Final Result
- Simon: [tex]\(1\)[/tex] (indicating no decrease in speed with increasing time)
- Raphael: [tex]\(1\)[/tex] (indicating a decrease in speed with increasing time)
So, the final answer is:
[tex]\[ (1, 1) \][/tex]
### Simon's Scenario
Simon drives faster as he enters the freeway from the entrance ramp. We are given the following data for Simon’s driving:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time} & \text{Speed} \\ \hline 0 & 0 \\ \hline 2 & 15 \\ \hline 4 & 25 \\ \hline 6 & 45 \\ \hline 8 & 70 \\ \hline \end{array} \][/tex]
To determine if Simon’s speed is decreasing with increasing time, we need to look at the changes in speed over time. We calculate the speed difference between consecutive times:
- From [tex]\( 0 \)[/tex] to [tex]\( 2 \)[/tex] seconds: [tex]\( 15 - 0 = 15 \)[/tex]
- From [tex]\( 2 \)[/tex] to [tex]\( 4 \)[/tex] seconds: [tex]\( 25 - 15 = 10 \)[/tex]
- From [tex]\( 4 \)[/tex] to [tex]\( 6 \)[/tex] seconds: [tex]\( 45 - 25 = 20 \)[/tex]
- From [tex]\( 6 \)[/tex] to [tex]\( 8 \)[/tex] seconds: [tex]\( 70 - 45 = 25 \)[/tex]
These differences indicate that Simon's speed is increasing over time because all the differences are positive.
To verify if the increments themselves are increasing, we review the differences between consecutive increments:
- From [tex]\( 15 \)[/tex] to [tex]\( 10 \)[/tex]: This decreases, which violates the condition for consistent increment.
- From [tex]\( 10 \)[/tex] to [tex]\( 20 \)[/tex]: This increases.
- From [tex]\( 20 \)[/tex] to [tex]\( 25 \)[/tex]: This increases.
Since there is an initial decrease in increments (from [tex]\( 15 \)[/tex] to [tex]\( 10 \)[/tex]), it's not a consistently increasing pattern of increments. But since all individual speed figures are increasing, Simon's speed indeed increases over time.
### Raphael's Scenario
Raphael rolls his ball downhill. When a ball rolls downhill without much or any control, typically its speed can vary depending on multiple factors like slope and friction. However, for the simplicity of this problem, we'll assume that "rolling downhill" implies that Raphael's speed is consistently decreasing with time as controlling speed may actually decrease due to various factors.
### Conclusion
Based on the provided prompt and analysis, here is the appropriate conclusion for each agent:
- For Simon's driving scenario, his speed increases over time rather than decreases.
- For Raphael's rolling ball scenario, it is reasonable to conclude that his speed decreases over time.
Thus, the correct representation corresponding to a decreasing speed with increasing time is for Raphael.
### Final Result
- Simon: [tex]\(1\)[/tex] (indicating no decrease in speed with increasing time)
- Raphael: [tex]\(1\)[/tex] (indicating a decrease in speed with increasing time)
So, the final answer is:
[tex]\[ (1, 1) \][/tex]
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