Problem 190 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

Yes, you can be cited for speeding as your average speed was 65 mph.

Step 1: Calculate Travel Time

First, determine how much time it took to travel between the two police cars. From 10:17 a.m. to 10:53 a.m. is a duration of 36 minutes.

Step 2: Convert Time to Hours

Convert the travel time into hours, as speed is measured in miles per hour. There are 60 minutes in an hour, so 36 minutes is \( \frac{36}{60} = 0.6 \) hours.

Step 3: Calculate Average Speed

Calculate the average speed by dividing the distance traveled by the time taken. The distance is 39 miles, and the time is 0.6 hours, so the average speed is \( \frac{39}{0.6} = 65 \text{ mph} \).

Step 4: Compare Average Speed to Speed Limit

Compare your average speed to the speed limit of 60 mph. Since your average speed was 65 mph, which is greater than the speed limit, you were speeding.

Key Concepts

Understanding Speed Limits

Speed limits are a fundamental part of road safety regulations and help govern how fast vehicles can travel on different roads. These limits are typically set depending on the type of road, environmental conditions, and surrounding areas to ensure everyone follows a safe speed.

When driving, it’s crucial to respect the marked speed limits as they are established for safety reasons. Exceeding these limits can endanger not just the driver but also passengers and other road users.

• Speed limits vary by location and country.
• They can change based on road conditions, such as in school zones or construction areas.
• Not adhering to speed limits can result in legal penalties.

It's important to keep an eye on speed signs and adjust your speed accordingly. When a speed limit is surpassed, it often results in consequences such as fines or license suspensions.

Distance and Time Calculation

Calculating distance and time is essential for determining speed and keeping within speed limits. In our original exercise, you passed two police cars 39 miles apart in 36 minutes. This type of situation requires understanding how to calculate average speed using distance and time.

The relationship between distance, speed, and time is captured by the formula:\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

• "Distance" is measured in miles or kilometers.
• "Time" should be in hours for these calculations.
• "Speed" will typically be miles per hour (mph) or kilometers per hour (kph).

In this example, 36 minutes is converted to hours by dividing by 60, resulting in 0.6 hours. Using the formula, you can find the average speed: \( \frac{39 \text{ miles}}{0.6 \text{ hours}} = 65 \text{ mph} \).
This average speed can then be compared to the speed limit to conclude if any speeding occurred.

Understanding Speeding Violations

Speeding violations occur when a driver exceeds the legally allowable speed limit. These violations are often detected through road enforcement methods, such as police patrols, speed cameras, and radar guns.

Speeding can lead to various consequences. For example, it increases the risk of accidents and can result in penalties such as tickets or fines. This makes understanding and adhering to speed limits crucial.

• Fines for speeding can vary based on the severity of the violation.
• Repeated violations may lead to more serious penalties, including suspension of a driver's license.

In our exercise, despite the individual instances of driving at 55 mph, the calculated average speed was 65 mph, which exceeded the 60 mph speed limit. Thus, you could indeed be cited for a speeding violation. It's important to always monitor your speed, especially over long distances, and ensure you are compliant with traffic regulations.