Let's break this problem down step by step to determine the minimum speed Sue must drive to catch Bob before the bus departs the rest stop.

Key Information:

1. Bob's Bus Trip:

   • Departure from Chicago: 12:30 PM
   • Arrival at rest stop: 3:30 PM
   • Rest stop departure: 4:00 PM
   • Bob spends 30 minutes at the rest stop.

2. Sue's Realization:

   • Sue realizes Bob left his driver's license at home at 1:15 PM.
   • She must catch Bob before the bus leaves the rest stop at 4:00 PM.

Time Analysis:

• Time available for Sue to reach the rest stop:
  • The bus leaves the rest stop at 4:00 PM.
  • Sue realizes the problem at 1:15 PM.
  • Therefore, Sue has from 1:15 PM to 4:00 PM to reach the rest stop, which gives her: $4:00 \, \text{PM} - 1:15 \, \text{PM} = 2 \, \text{hours} \, 45 \, \text{minutes}$ In hours, this is: $2 + \frac{45}{60} = 2.75 \, \text{hours}$

Minimum Speed Calculation:

To determine the minimum speed Sue must drive, we need to know the distance between Chicago and the rest stop. Since this information is not provided in the problem, let's denote this distance as $d$ miles.

• Sue's minimum speed to cover this distance $d$ in the 2.75 hours available is given by the formula for speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ Substituting the time: $\text{Minimum speed} = \frac{d}{2.75}$

Conclusion:

• Sue needs to drive at a minimum speed of $\frac{d}{2.75}$ miles per hour (mph), where $d$ is the distance from Chicago to the rest stop.

If you know the distance $d$, you can plug it into the equation to find the exact speed. Do you know the distance, or would you like to explore other factors or assumptions?

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Would you like further details on any specific part of this solution? Here are 5 follow-up questions to consider:

1. What if Sue realized the mistake later, at 2:00 PM? How would the speed change?
2. How would a traffic delay impact Sue's chances of catching the bus?
3. What if Bob's rest stop time was longer (e.g., 1 hour instead of 30 minutes)?
4. If Sue had to drive through heavy traffic, how could that change her speed requirements?
5. What if the bus took a detour, adding distance to its route? How would that affect Sue's timing?

Tip: Always double-check time conversions when dealing with real-world problems to avoid small errors that can lead to large impacts in calculations!