Question

The table shows the distances y a motorcyclist is from home after x hours. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Time (hours), } \boldsymbol{x} & 0 & 1 & 2 & 3 \\ \hline \text { Distance (miles), } \boldsymbol{y} & 0 & 45 & 90 & 135 \\ \hline \end{array} $$

Short answer

The motorcyclist drives at a constant speed of 45 miles per hour.

Step-by-step solution

Identify the Independent and Dependent Variables

In the given data table, the independent variable x is the 'Time (hours)' and the dependent variable y is the 'Distance (miles)'.

Calculate the slope using two data points

It's possible to determine the slope of the line by using any two data points from the table. The formula for the slope (m) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For instance, using the points (1,45) and (2,90), this yields \( m = \frac{90 - 45}{2 - 1} = 45 \) miles per hour.

Interpret the slope

The slope of the line represents the rate of change of the distance with respect to time. Here, it's seen that the distance changes by 45 miles for every hour. Hence, the motorcyclist drives at a constant speed of \(45\) miles per hour.

Key concepts

Independent and Dependent Variables

Understanding the role of independent and dependent variables is crucial when analyzing data. In the context of the distance traveled by a motorcyclist over time, the independent variable is the 'Time (hours)', denoted as 'x'. This variable is independent because it can be chosen freely - we can pick any point in time to measure. On the other hand, the 'Distance (miles)', denoted by 'y', is the dependent variable because it relies on the amount of time that has passed, hence depending on the independent variable. As time progresses, the distance changes in response.

In our exercise, as time increases, the distance from home also increases, showing a direct relationship between these two variables. By recognizing this, we can start to predict the motorcycle's position at different times during its journey.

Rate of Change

The rate of change is a way of expressing how much one quantity, the dependent variable, changes for every incremental change in another quantity, the independent variable. Typically in a mathematical context, we see this as the slope of a line. For the motorcyclist, the rate of change is his speed because it shows how quickly his distance from home increases with every hour that passes.

By calculating the slope of 45 using the given data, we've determined that the motorcyclist's rate of change is 45 miles per hour. This means for every hour the motorcyclist travels, the distance from home changes by 45 miles — a relatively consistent and easy-to-understand concept of movement over time.

Linear Relationships

A linear relationship is one where the rate of change between two variables is constant, which means the dependent variable changes by a fixed amount for each unit change in the independent variable. In a graph, this relationship is represented as a straight line.

Given the data table for the motorcyclist's distance over time, we can graph the points and draw the line that connects them. This line will show a consistent, straight path, highlighting that the relationship between time and distance is linear. The consistent speed of 45 miles per hour confirms that for every additional hour traveled, the motorcyclist is consistently 45 miles farther from the starting point.

Interpreting Slope

Interpreting the slope is about understanding what the value represents in real-world terms. In our example, the slope is calculated to be 45, signifying that for every unit of time passed (in this case, every hour), the distance from home increases by 45 miles.

Therefore, the slope tells us the travel speed of the motorcyclist. It is an indicator of his pace and helps us understand how quickly he is moving away from or potentially towards a certain point. It's a powerful number that summarizes his consistent rate of travel over time.

Distance-Time Relationship

The distance-time relationship is a way to describe how distance changes as time progresses. For the motorcyclist, this relationship indicates that the farther he travels, the longer he has been on the road, given that he maintains a steady speed.

Picturing this relationship on a graph, with time on the horizontal axis and distance on the vertical axis, forms a line that rises to the right. This ascending line is a visual representation of the journey, showing how over each hour the motorcyclist is consistently 45 miles further from where the journey began. Not only does it tell us the speed but it also enables us to forecast the future position or to reconstruct past locations at specific times.