Problem 50 - Thomas Calculus in SI Units

Question

A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered \(230 \mathrm{km}\) on a toll road with speed limit \(100 \mathrm{km} / \mathrm{h}\). The trucker was cited for speeding. Why?

Step-by-Step Solution

Answer

The trucker's average speed was 115 km/h, exceeding the speed limit of 100 km/h.

Step 1: Understand the Problem

The trucker traveled 230 km in 2 hours. We need to find out the average speed during the travel and compare it with the speed limit of 100 km/h to determine if the trucker was speeding.

Step 2: Calculate Average Speed

The average speed can be calculated using the formula: \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \). Here, \( \text{Total Distance} = 230 \) km and \( \text{Total Time} = 2 \) hours. So, \( \text{Average Speed} = \frac{230}{2} = 115 \) km/h.

Step 3: Compare with Speed Limit

The calculated average speed of 115 km/h is greater than the road's speed limit of 100 km/h.

Step 4: Conclusion

Since the average speed of the trucker was 115 km/h, which exceeds the speed limit of 100 km/h, this is why the trucker was cited for speeding.

Key Concepts

Speed Limit

The speed limit is a crucial aspect of road safety measures. It sets the maximum legal speed at which vehicles can travel on a particular road.
Speed limits are established for different reasons:

• Safety: To reduce the risk of accidents, especially in areas with high pedestrian activity or challenging driving conditions.
• Road conditions: Considering the design and condition of the roads.
• Environmental concerns: To control vehicle emissions and noise levels.

In the example exercise, the speed limit on the toll road was set at 100 km/h. This means that any vehicle traveling faster than that limit may be cited for speeding, as was the case with the trucker. This enforcement helps ensure that all drivers maintain a pace that is considered safe for the road's specifics.

Distance Formula

The distance formula is an essential tool in calculating the distance traveled during a trip. It is simply the relationship between the total distance traveled, time taken, and speed.
In mathematical terms, this can be represented as:\[\text{Distance} = \text{Speed} \times \text{Time}\]When you know any two of these values, you can easily find the third. For instance, if you know the speed and the time, you can find the distance.
For the problem with the trucker, we already have the total distance of 230 km covered in 2 hours. By rearranging the formula, if needed, you can find any one of these three values based on the information available.

Time-Speed-Distance Relationship

Understanding the relationship between time, speed, and distance is fundamental for solving many real-world problems involving travel.
Here are the basic principles:

• Speed is defined as how fast an object is moving. It is usually expressed in terms of distance per unit of time, such as km/h or m/s.
• Time is the period over which the travel has occurred.
• The distance is the total length of the journey between two points.
  

If you want to find the average speed over a journey, use the formula:\[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\]In the trucker's scenario, this formula allows us to calculate that with a 230 km trip over 2 hours, the average speed was 115 km/h. Comparing this with the speed limit indicated why the trucker was cited, since exceeding the speed limit posed safety concerns.