Problem 46 - Thomas Calculus

Question

A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?

Step-by-Step Solution

Answer

The trucker's average speed was 79.5 mph, which exceeds the 65 mph speed limit; hence, she was cited for speeding.

Step 1: Determine the average speed

To find out why the trucker was cited for speeding, we first need to calculate the average speed. The formula for average speed is \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]. In this case, the total distance is 159 miles and the total time is 2 hours. Thus, the average speed is \[ \frac{159}{2} = 79.5 \text{ mph} \].

Step 2: Compare the average speed to the speed limit

The speed limit on the toll road is given as 65 mph. We compare the trucker's average speed of 79.5 mph to the speed limit. Since 79.5 mph is greater than 65 mph, the trucker was exceeding the speed limit.

Step 3: Conclusion

Since the trucker's average speed of 79.5 mph is above the speed limit of 65 mph, the trucker was correctly cited for speeding.

Key Concepts

Speed Limit Comparison

When driving on roads, it's crucial to adhere to the posted speed limits. These limits are set to ensure safety for all road users. In this trucker's case, we calculated her average speed to be 79.5 mph over a 2-hour drive.

• Speed Limit: 65 mph
• Trucker's Average Speed: 79.5 mph

Comparing the two speeds, the trucker's average is significantly higher than the speed limit. This clearly shows that the trucker was not adhering to the traffic regulations regarding speed limits. Not only does this pose a risk to the driver but also to others on the road. Driving over the speed limit can reduce reaction time and increase the severity of accidents. Therefore, the citation for speeding was warranted.

Distance-Time Relationship

The relationship between distance and time is a fundamental concept in understanding speed. In this scenario with the trucker, we consider the formula: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
By re-arranging and substituting into this formula, we can clearly determine how the trucker managed such high speeds. Driving 159 miles in only 2 hours indicates the trucker traveled consistently fast. This information is not just crucial for identifying speed limit violations but also helps in planning long trips by predicting arrival times.

• If speed is too high, it indicates rushing and possible rule violations.
• If speed is within limits, it shows compliance with regulations and often results in safer driving conditions.

Traffic Regulations

Traffic regulations are rules that govern how drivers should behave on roads to ensure everyone’s safety. One of the most critical aspects of these regulations is speed limits. These limits are established based on road conditions, traffic density, and safety research. In this scenario, disregarding the speed limit of 65 mph sets a precedent for unsafe driving. Not only do these rules aim to keep the driver safe, but they are also in place to protect pedestrians and other drivers.

• Speeding increases the risk of accidents.
• Increased speed reduces the time to react to hazards.
• Consistently speeding can lead to multiple violations and penalties.

Adhering to traffic regulations is not only about avoiding fines but also about being responsible on the road. Safe driving practices, including adhering to speed limits, reduce the likelihood of traffic mishaps.