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Your Company plans to transport 20 horses to a racetrack in Denver. It takes 12 hours to drive directly from Phoenix to Denver. You must stop to feed the horses every 2 hours and need a six-hour sleep every 10 hours of driving. If you are responsible for this move with only 2 days (48 hours) left to meet your deadline, and only considering feeding the horses and your own sleep requirements, what is your critical ratio?
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The problem involves calculating the critical ratio relevant to the logistics of transporting horses from Phoenix to Denver within a specified timeframe, considering mandatory stops for feeding and sleep. A critical ratio typically compares the available time against the total required time, reflecting the urgency or risk associated with meeting the deadline.
First, let's understand the parameters:
- Driving time from Phoenix to Denver: 12 hours
- Feeding stops are required every 2 hours of driving
- Sleep must be taken every 10 hours of driving, with a 6-hour sleep period each time
- Total available time to meet the deadline: 48 hours (2 days)
Our goal is to calculate the total time needed to complete the trip, including all mandatory stops, within the deadline. The critical ratio (CR) is then calculated as:
CR = (Available Time) / (Total Required Time)
which indicates how the necessary time compares to what's available. A CR of 1 or higher suggests feasible completion; less than 1 indicates a risk of not meeting the deadline.
Step 1: Determine traveling time
Base driving time: 12 hours.
Step 2: Calculate feeding stops
During the 12-hour drive, feeding stops occur every 2 hours.
Number of feeding stops = total driving time / time between feeds = 12 / 2 = 6 stops.
Assuming each feeding stop takes negligible time relative to driving, but in reality, they do require some time. However, since no specific duration for feeding is given, we will assume the effective impact is covered by the stops themselves being counted as stops without adding extra time—it is implied feeding time is significant but not quantified here. If needed, one could assume a fixed time per feeding stop; typical feeding may take approximately 15 minutes per stop.
For a conservative estimate, let's assume 15 minutes (0.25 hours) per feeding stop. Total feeding time = 6 stops * 0.25 hours = 1.5 hours.
Step 3: Calculate sleep stops
Sleep is required every 10 hours of driving, with a 6-hour sleep period each time.
Total hours of driving: 12 hours.
Number of sleep periods needed: since sleep is needed after every 10 hours, and total driving is 12 hours, only 1 sleep period is necessary (because after 10 hours, sleep is required, but driving only continues to 12 hours, which is less than two sets of 10 hours).
Therefore, total sleep time = 6 hours.
Step 4: Total time calculation
Sum of all time components:
- Driving time: 12 hours
- Feeding time: 1.5 hours
- Sleep time: 6 hours
Total estimated time = 12 + 1.5 + 6 = 19.5 hours.
Step 5: Calculate the critical ratio
The available time is 48 hours; the total required time is approximately 19.5 hours. Therefore,
CR = 48 / 19.5 ≈ 2.46
This critical ratio of approximately 2.46 indicates that, under current assumptions, the trip can be completed comfortably within the 48-hour deadline with plenty of time remaining, considering only feeding and sleep constraints.
Additional considerations
It is essential to note that the assumptions here are simplified. If actual feeding or rest times are greater, or if unexpected delays occur, the total required time may increase. The critical ratio would then decrease, reflecting increased risk. However, based on the given data, the process appears feasible well within the deadline.
Conclusion
The critical ratio, calculated as the ratio of available time to required time considering feeding and sleep needs, is approximately 2.46. This suggests a comfortable margin to meet the 2-day deadline for the transportation of horses from Phoenix to Denver under the specified constraints.
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