Problem 50 - 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 speed limit of 65 mph.

Step 1: Understand the Problem

We are asked to determine if the trucker was speeding based on the distance covered and the time taken. The given speed limit is 65 mph.

Step 2: Calculate Average Speed

To find the trucker's average speed, use the formula: \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \). The total distance covered is 159 miles, and the total time is 2 hours. So, \( \text{Average Speed} = \frac{159}{2} = 79.5 \text{ mph} \).

Step 3: Compare with Speed Limit

Now compare the calculated average speed, 79.5 mph, with the speed limit, which is 65 mph. Since 79.5 mph is greater than 65 mph, the trucker was indeed speeding.

Key Concepts

Understanding Average Speed

Average speed is a concept used to determine how fast, on average, an object is moving over a period of time. It is calculated by dividing the total distance traveled by the total time taken to travel that distance. This simple formula is expressed as: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]For example, if a vehicle covers 159 miles in 2 hours, its average speed is calculated as 79.5 mph. This means, on average, the vehicle was moving at a speed of 79.5 miles per hour over the entire journey.

• Average speed helps in understanding overall movement efficiency.
• It's important when evaluating adherence to speed regulations.
• Different from instantaneous speed, which is the speed at any single point in time.

Recognizing the Importance of Speed Limits

Speed limits are put in place to enhance road safety by setting maximum speeds at which vehicles may legally travel in certain areas. These limits are crucial because they:

• Ensure the safety of all road users by reducing the risk of accidents.
• Help manage traffic flow smoothly and efficiently, especially on busy roads.
• Promote fuel efficiency and reduce pollution.

In the given problem, the speed limit was 65 mph on the toll road. By traveling at an average speed of 79.5 mph, the trucker exceeded the speed limit, thereby being cited for speeding. It's critical to adhere to speed limits not just for legal reasons, but for the safety and well-being of everyone on the road.

Exploring the Distance-Time Relationship

The distance-time relationship is a key concept in understanding how motion is quantified. Distance and time are directly related to speed:

• If distance increases and time stays the same, speed increases.
• If distance stays the same while time increases, speed decreases.

In our problem, the trucker drove 159 miles in 2 hours, which relates to her average speed of 79.5 mph. By comparing this calculated speed with the legal speed limit, we can determine whether the trucker's driving adhered to traffic laws.
Understanding this relationship is crucial in various scenarios, from simple travel planning to more complex physics applications. It plays a key role in day-to-day traffic management and personal travel decisions. Being able to predict travel time or required speed for a trip helps in planning journeys, adhering to laws, and ensuring safety. Remember, the balance between distance, time, and speed is essential in motion-related calculations.