Problem 190 - Calculus Volume 1

Question

At 10:17 a.m., you pass a police car at 55 mph that is stopped on the freeway. You pass a second police car at 55 mph at 10:53 a.m., which is located 39 mi from the first police car. If the speed limit is 60 mph, can the police cite you for speeding?

Step-by-Step Solution

Answer

Yes, you were exceeding the speed limit by 5 mph.

Step 1: Calculate Time Spent Driving

First, find the time interval between passing the first police car and the second police car. Calculate the time difference: from 10:17 a.m. to 10:53 a.m., which is 36 minutes.

Step 2: Convert Time to Hours

Since speed is typically measured in miles per hour (mph), convert the time spent driving into hours. 36 minutes is equivalent to \( \frac{36}{60} = 0.6 \) hours.

Step 3: Calculate Average Speed

To get the average speed, use the formula: \( \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \). You traveled 39 miles in 0.6 hours: \[ \text{Average speed} = \frac{39}{0.6} \approx 65 \text{ mph} \].

Step 4: Compare with Speed Limit

The average speed calculated is 65 mph. Compare this with the speed limit of 60 mph. Since 65 mph is greater than 60 mph, you were exceeding the speed limit.

Key Concepts

Time Calculation

Understanding how to calculate time is crucial when you are assessing speed and distance. In order to estimate your travel time correctly, you need to look at the clock carefully and use basic arithmetic to determine how long your journey takes.

Here's a step-by-step guide on how to perform this calculation:

• First, note the exact time you begin your travel, like when you pass the first location or start point. In our example, this was 10:17 a.m.
• Next, record the time when you reach the second point or the endpoint, which was 10:53 a.m. in this scenario.
• Subtract the start time from the end time to find the total minutes traveled. The difference was 36 minutes in this case.
• To make it compatible with the standard 'miles per hour' measurement, convert this travelling time into hours, which simplifies calculations and comparisons. This can be done by dividing the minutes by 60 (the number of minutes in an hour). So, in our example, 36 minutes equals 0.6 hours.

Speed Limit

The speed limit is a legal boundary placed on vehicles to ensure safety on the roads. It is a maximum allowable speed that you can travel at. Always adhere to the speed limit to avoid fines and ensure safety. In this problem, the speed limit on the freeway is given as 60 miles per hour (mph).

Understanding and sticking to the speed limit is not just about avoiding citations. It helps you:

• Maintain control of the vehicle especially in tricky conditions.
• Ensure the safety of yourself and others.
• React quickly to unforeseen road hazards.

If you calculate your average speed and it surpasses the speed limit, like in our example where the average speed was 65 mph, you know that you are driving faster than legally allowed, which is risky and punishable.

Distance and Time Relationship

Understanding the relationship between distance and time is essential for computing average speed, which in turn, informs whether you're adhering to speed regulations. In essence, speed is how fast or slow an object, such as a car, moves across a particular distance over a given time period.

To express this relationship, the formula \[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \] can be extremely helpful. When you break down this formula:

• Total distance: This is how far you've traveled. In the given problem, it's 39 miles.
• Total time: This is the duration for which you've been traveling. This was calculated as 0.6 hours in the problem.
• Using these two components, you can find the average speed by dividing the distance by time.

Understanding this relationship helps in determining whether you are speeding or not. With an average speed of 65 mph calculated in the earlier solution, you now know that you exceeded the allowable limit of 60 mph. This clarifies why you might be getting cited for speeding.