Problem 52 - University Calculus: Early Transcendentals

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 was cited for speeding because her average speed was 79.5 mph, which exceeds the 65 mph speed limit.

Step 1: Calculate Average Speed

To find out if the trucker was speeding, we first need to calculate their average speed over the 2-hour trip. The average speed can be calculated using the formula: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]In this case, the total distance is 159 miles, and the total time is 2 hours. Plugging in these values gives:\[ \text{Average Speed} = \frac{159}{2} = 79.5 \text{ mph} \]

Step 2: Compare with Speed Limit

Now, we need to compare the average speed calculated in Step 1 with the given speed limit. The speed limit is 65 mph. We calculated the trucker’s average speed as 79.5 mph.

Step 3: Determine If Speeding Occurred

Since the trucker's average speed of 79.5 mph is greater than the speed limit of 65 mph, she was exceeding the speed limit. This justifies the citation for speeding.

Key Concepts

Average Speed Calculation

Understanding how to calculate average speed is key to solving many real-world problems, such as determining whether a driver was speeding. Average speed represents the total distance traveled divided by the total time taken.
For example, if a truck driver travels 159 miles in 2 hours, you would calculate the average speed with this formula: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \] When you insert the values specific to this scenario: \[ \text{Average Speed} = \frac{159}{2} = 79.5 \text{ mph} \] This value shows how fast, on average, the driver was traveling over this 2-hour period.

• The time unit in both the numerator and the denominator must match.
• The formula gives an average, meaning speed can vary during the journey.
• Simpler than calculating each portion of a mixed-speed journey separately.

Recognizing that average speed does not account for variables such as stops or changes in pace is vital. This concept simplifies a potentially complex trip into a straightforward calculation.

Speed Limit Comparison

Once the average speed is determined, comparing it to the applicable speed limit is straightforward. This comparison helps determine compliance with road regulations.
In this context, recognizing whether a speeding violation happened involves evaluating if the calculated average speed is greater than the speed limit set for the road.
Given that the trucker's average speed was 79.5 mph and the speed limit stood at 65 mph:

• If 79.5 mph > 65 mph, as in this example, speeding has occurred.
• Conversely, if the average speed is less than or equal to the speed limit, no violation takes place.

This simple comparison checks adherence to speed laws and is vital for maintaining road safety. It’s worth noting that the average speed exceeding the limit even slightly can result in a citation, making precision in these calculations important for legal reasons.

Distance and Time Calculation

At the heart of speed analysis are the concepts of distance and time calculation, essential for understanding travel and movement.
Distance refers to the total length of the travel path, often measured in miles or kilometers. Time is the duration required to traverse this distance.
When working with speed-related problems:

• Always confirm the units are consistent; miles with hours, kilometers with hours, etc.
• Ensure accurate measurement of time intervals to avoid skewed results.

In practical scenarios, like toll booth analysis or traffic monitoring, correctly understanding these elements allows for effective speed management and regulation adherence. Proper measurement and consistency in units ensure effective analysis of average speeds against speed limits, underscoring the importance of careful calculation in transportation and logistics. Keep in mind that minor measurement errors can lead to significant differences, especially when dealing with legal regulations regarding speed limits.