Problem 1 - Precalculus Mathematics for Calculus

Question

If you travel 100 miles in two hours, then your average speed for the trip is average speed \(=\) _____ = _____.

Step-by-Step Solution

Answer

Average speed is 50 miles per hour.

Step 1: Understand the formula for Average Speed

The average speed is calculated by dividing the total distance traveled by the total time taken. The formula is: \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \).

Step 2: Plug in the Total Distance

You are given that the total distance traveled is 100 miles. So in the formula, replace 'Total Distance' with 100: \( \text{Average Speed} = \frac{100}{\text{Total Time}} \).

Step 3: Plug in the Total Time

You are given that the total time taken is 2 hours. So in the formula, replace 'Total Time' with 2: \( \text{Average Speed} = \frac{100}{2} \).

Step 4: Calculate the Average Speed

Divide 100 by 2 to find the average speed: \( \text{Average Speed} = 50 \text{ miles per hour} \).

Key Concepts

Distance-Time Relationship

The relationship between distance and time is one of the foundations of understanding motion. As you travel, the distance you cover over a specific period can help determine how fast you are moving on average. Imagine it like capturing a snapshot of the entire journey in terms of how far you've gone and how long it took. This concept is critical because:

• It helps in planning trips efficiently by predicting how long a journey will take.
• It allows comparison of the speed between different trips or vehicles.

In our example, traveling 100 miles in 2 hours puts you on a consistent path where each hour sees you advancing 50 miles. This is evident by the formula for average speed: \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \).
Understanding this relationship helps us comprehend traffic and travel dynamics better.

Ratio and Proportion

Understanding ratios and proportions is essential to solve problems involving distance, speed, and time. The ratio of distance to time gives us the concept of speed. Speed itself is a proportional relationship between how much distance is covered per each unit of time. This can be visualized by imagining that as the distance doubles, the time doubles if the speed remains constant.
In the calculation of average speed, we are applying this ratio to determine:

• The consistency of speed throughout a journey.
• Adjustments needed if the time or distance changes.

So when you see that traveling 100 miles in two hours results in an average speed of 50 miles per hour, you are directly observing this ratio in action: \( \text{Average Speed} = \frac{100}{2} \).
This consistent measurement allows for clear planning and adjustments in travel scenarios.

Unit Conversion

Unit conversion is crucial when working with measurements since it ensures accuracy and understanding across different systems. Whether it's miles to kilometers or hours to minutes, knowing how to convert units properly is essential for solving problems correctly.
In our exercise, we're working with miles and hours. If the problem had been given in kilometers and minutes, we would need to:

• Convert miles to kilometers (1 mile ≈ 1.609 km).
• Convert hours to minutes (1 hour = 60 minutes).

This conversion helps ensure that comparisons are consistent and understandable. It's vital to always check if conversions are necessary when faced with unfamiliar units. Implementing accurate unit conversion ensures that when you calculate an average speed of 50 mph from 100 miles in 2 hours, it can smoothly translate to other units if needed.