Problem 146 - Calculus Volume 1

Question

To get on a certain toll road a driver has to take a card that lists the mile entrance point. The card also has a timestamp. When going to pay the toll at the exit, the driver is surprised to receive a speeding ticket along with the toll. Explain how this can happen.

Step-by-Step Solution

Answer

The toll system calculates average speed from entry and exit data; if it exceeds the speed limit, a speeding ticket is issued.

Step 1: Understand the System

When entering the toll road, the driver takes a card that records their entry point in miles and a timestamp noting the exact time of entry.

Step 2: Analyze the Exit Process

Upon exiting, the toll system captures the end mile point and the timestamp of when the driver leaves the toll road.

Step 3: Calculate Average Speed

The toll system uses the entry and exit times, as well as the distance traveled, to calculate the average speed using the formula:\[\text{Average Speed} = \frac{\text{Distance Traveled (miles)}}{\text{Time Taken (hours)}}\] This is calculated from the difference in mile markers and time stamps.

Step 4: Determine Speeding

Compare the calculated average speed to the posted speed limit of the toll road. If the average speed exceeds the speed limit, the system flags the vehicle for speeding.

Step 5: Issue a Speeding Ticket

If speeding is detected, the toll system issues a speeding ticket along with the toll charge.

Key Concepts

Understanding Average Speed Calculation

Average speed is a simple yet crucial concept in understanding how speed is monitored and enforced. It's calculated by dividing the total distance traveled by the total time taken for that journey. Here's the formula: \[\text{Average Speed} = \frac{\text{Distance Traveled (miles)}}{\text{Time Taken (hours)}}\]Breaking this down further:

• Distance Traveled: This is the difference in mile markers from where the journey started to where it ended. It's measured in miles.
• Time Taken: The time difference between when the driver enters the toll road and when they exit. It's usually measured in hours.

Calculating average speed helps in understanding overall journey speed but doesn't tell us about the speed at particular instances. It's possible a driver went slowly for some parts and fast for others but still ends up with an average speed that’s above the limit if they spent more time speeding in certain intervals.

Speed Limit Enforcement Through Tolls

Toll roads are increasingly using technology to enforce speed limits. Here's how the process typically works: When you enter a toll road, the toll system records your entry point (in miles) and the exact time you start your journey. When you exit, the system records the ending mile point and the exit time. With these data points, the toll system can calculate your average speed. Speed limits are set to ensure safety on the roads. If a vehicle's average speed exceeds the set limits, it triggers an alert for potential speeding. The system automatically issues a speeding ticket along with the toll charge if a violation is detected. The advantage of using toll enforcement is that it doesn’t rely solely on physical police presence, but rather on a calculated method that captures overall compliance over the duration of the trip. This method reduces human error and increases efficiency in maintaining road safety.

Exploring Distance-Time Relationship

The relationship between distance and time is fundamental in understanding motion and speed monitoring on roads. At its core, this relationship is about how far an object travels over a given period.

• Distance: Refers to the length of the path traveled by the vehicle between entry and exit points.
• Time: Represents the total duration taken to complete the journey.
• Speed: When distance and time are known, speed measures how fast a vehicle is traveling, calculated by dividing distance by time.

In practical terms, understanding this relationship helps road systems like toll roads accurately assess whether a vehicle has been speeding. By calculating the ratio of distance to time, toll systems effectively measure motion over the period, ensuring drivers adhere to designated safe speed limits. This relationship is a cornerstone of physics and is crucial for applying theoretical concepts to real-world situations.