Problem 8 - 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}$$ Given \(t,\) the time in hours, write an equation that can be used to find \(d\), the distance driven.

Step-by-Step Solution

Answer

The equation for the distance driven is \(d = 55t\).

Step 1: Understand the Variables

Identify the two variables in the problem: \(t\), which represents time in hours, and \(d\), which represents distance in miles.

Step 2: Recognize the Relationship

Notice from the table that as time \(t\) increases, the distance \(d\) also increases proportionally, indicating a linear relationship.

Step 3: Determine the Rate

From the problem, the family drives at an average speed of 55 miles per hour. This means for every 1 hour, they travel 55 miles.

Step 4: Construct the Equation

The relationship between distance and time for a constant speed can be described by the equation \(d = rt\), where \(r\) is the rate or speed. Here, \(r = 55\) miles per hour, so the equation becomes \(d = 55t\).

Key Concepts

Distance and Time Relationship

The relationship between distance and time in motion scenarios is a foundational concept in physics and mathematics. In this context, the two main variables are time, typically represented as \(t\), and distance, represented as \(d\). Understanding how these variables interact helps us predict how far something will travel over a period.

In our exercise, time \(t\) is measured in hours, and distance \(d\) is measured in miles. Key to this relationship is the idea that as time passes (increases), the total distance traveled also increases. This implies that time and distance have a direct and proportional relationship.

Consider this simple example: If a car travels at a steady speed of 55 miles per hour, after 1 hour, the car covers 55 miles. After 2 hours, the car covers 110 miles, doubling its distance. This consistent rate of travel demonstrates the steady increase in miles traveled over consistent time intervals, confirming our understanding of the distance and time relationship.

Average Speed

Average speed is an essential concept when analyzing motion and movement. In simple terms, it tells us how much distance is covered in a given amount of time. It is calculated using the formula:

\[\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}\]
In our exercise, the average speed is given as 55 miles per hour, indicating that the Geiger family travels 55 miles every hour.

Key points about average speed:

• The average speed tells us about the velocity of an object over the total time, not just at a specific moment.
• When speed is constant, it makes the calculations straightforward, as in our exercise.
• Average speed is crucial for planning trips since it helps us estimate how long a journey will take or how far we can go within a specific time.

Understanding average speed allows us to construct meaningful equations, such as the one given in the exercise, \(d = 55t\), which directly links the time traveled to the distance covered.

Linear Relationship

A linear relationship refers to a scenario where two quantities have a constant rate of change. In mathematical terms, this means the graph of these two quantities forms a straight line.

In the exercise, the linear relationship is observed between time (\(t\)) and distance (\(d\)). This relationship can be expressed with a simple linear equation:
\[d = rt\]Here, \(r\) represents the constant rate, or the speed, which is 55 miles per hour in our example. The equation showcases a direct proportionality:

• For each additional hour (\(t\)), there is a proportional increase in distance (\(d\)).
• The graph of this equation would appear as a straight line passing through the origin (0,0), reflecting the constant speed.

Recognizing linear relationships is beneficial in predicting and understanding how two variables interact, enabling us to use simple algebraic expressions to model the behavior of real systems, such as travel and motion.