Problem 9 - Prealgebra

Question

The Geiger family is driving at an average speed of 55 miles per hour. The table shows the relationship between the distance driven and the time. $$\begin{array}{|c|c|}\hline \text { Time \(t\) } & \text { Distance \(d\) } \\\\\hline \text { (hours) } & \text { (miles) } \\\\\hline 1 & 55 \\\\\hline 2 & 110 \\\\\hline 4 & 220 \\\\\hline 5 & 275 \\ \hline\end{array}$$ How long would it take them to drive 495 miles?

Step-by-Step Solution

Answer

It takes 9 hours to drive 495 miles.

Step 1: Understand the Problem Context

The problem involves a scenario where the Geiger family is driving at a constant speed of 55 miles per hour. We have to determine how long it will take them to drive 495 miles.

Step 2: Identify the Given Information

The average speed is given as 55 miles per hour. We need to determine the time it takes to travel a known distance, which is 495 miles.

Step 3: Recall the Relationship Between Distance, Speed, and Time

The formula to calculate time when speed and distance are known is given by: \[ t = \frac{d}{s} \]where \(t\) is the time, \(d\) is the distance, and \(s\) is the speed.

Step 4: Substitute the Known Values into the Formula

Using the formula from Step 3, substitute \(d = 495\) miles and \(s = 55\) miles per hour:\[ t = \frac{495}{55} \]

Step 5: Perform the Calculation

Carry out the division of 495 by 55 to find the time:\[ t = 9 \]Thus, it takes 9 hours to drive 495 miles at a constant speed of 55 miles per hour.

Key Concepts

Average Speed

Understanding the concept of average speed is crucial when dealing with problems involving distance, speed, and time relationships. Average speed is defined as the total distance traveled divided by the total time taken. It describes how fast an object is moving overall, regardless of any variations in speed during the journey. For instance, if the Geiger family travels 495 miles in 9 hours, their average speed would be calculated as \( \text{Average Speed} = \frac{495 \text{ miles}}{9 \text{ hours}} = 55 \text{ mph}\).
This value, 55 mph, indicates their constant or typical speed throughout the entire duration of the trip. Knowing the average speed helps us to predict future travel time when covering other long distances like in the Geiger family example. Whenever the speed remains consistent each hour, it simplifies calculations, hence emphasizing the importance of understanding average speed.

Unit Conversion

Unit conversion is an essential skill needed to solve problems in mathematics and physics. When dealing with the distance-speed-time relationship, ensuring all units match is important for accuracy.
The most common units of speed are miles/hour or kilometers/hour. When presented with a problem like the Geiger family, where the speed is given in miles per hour, ensure the distance is also measured in miles to avoid conversion mistakes. For situations where different units are involved, like converting kilometers to miles, use the conversion factor: 1 mile is approximately 1.60934 kilometers.

• To convert miles to kilometers: multiply by 1.60934.
• To convert kilometers to miles: divide by 1.60934.

Staying on top of unit conversions helps prevent inaccuracies in problem solving and provides a universal approach to interpreting speed and distance calculations.

Problem Solving

Approaching problem-solving in the context of distance-speed-time relationships involves several structured steps. This ensures solutions are both logical and accurate.
To tackle such problems, begin by carefully understanding the problem and identifying what is given and what needs to be found. In our example, we know the Geiger family's average speed (55 mph) and the required distance to travel (495 miles).
Recall the relationship between distance, speed, and time using the formula \( t = \frac{d}{s} \). This relationship acts as the foundation for calculating time when either speed or distance is given. By substituting known values into the formula, you can simplify the expression to find the unknown variable effectively. Finally, perform the calculation, interpreting the result in the context of the problem.
These systematic steps help organize various aspects of the problem, ensuring that you address each part thoroughly and arrive at the correct solution.