Problem 32 - Algebra for College Students

Question

Suppose that Bianca walks at a constant rate of 3 miles per hour. Explain what it means that the distance Bianca walks is a linear function of the time that she walks.

Step-by-Step Solution

Answer

Bianca's distance is a linear function of time because she walks at a constant rate, forming a straight line when graphed.

Step 1: Define Linear Function

A linear function is a function of the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. The function graph is a straight line.

Step 2: Interpret Distance as a Linear Function

Let \( d(t) \) be the distance Bianca walks as a function of time \( t \). Since she walks at a constant rate, \( d(t) = rt \) where \( r = 3 \) miles/hour is the rate. This equation fits the form \( f(x) = mx + b \) with \( m = 3 \) and \( b = 0 \).

Step 3: Identify the Components of the Linear Function

In the function \( d(t) = 3t \), the slope \( m = 3 \) represents Bianca's walking rate, meaning for every hour she walks, she covers 3 miles. The y-intercept \( b = 0 \) suggests that at time \( t = 0 \), she has walked 0 miles.

Step 4: Explain the Implications

Because distance is a linear function of time at a constant rate, the distance Bianca walks increases at a steady rate without variation. The graph of \( d(t) = 3t \) is a straight line passing through the origin, consistent with a constant rate of change.

Key Concepts

Constant Rate

When we talk about a constant rate, we refer to a situation where something changes by the same amount over equal intervals of time. In Bianca's case, she walks 3 miles every hour. This means that for each hour that passes, she covers exactly 3 more miles, regardless of how long she has been walking.

In mathematical terms, a constant rate is expressed by a fixed number that describes how quickly something happens. For Bianca, the rate of walking is 3 miles per hour. Here, there is no change in speed; she doesn't speed up or slow down.

Constant rates are common in many daily activities, like driving a car at a steady speed or filling a tub with water at a uniform pace. In the world of mathematics and physics, understanding constant rates helps in predicting and modeling how things change with time. They make calculations straightforward and establish a reliable relationship, much like Bianca's predictable walking pattern.

Graph of a Function

The graph of a function is a visual way to represent how the values of a function change. For Bianca's walking, the graph shows the relationship between the time she walks and the distance she covers.

If we were to graph the function describing her walk, we would place time (in hours) on the x-axis and distance (in miles) on the y-axis.
The graph of the function \( d(t) = 3t \) is a straight line, starting at the origin (0,0) because at time zero, she has walked zero miles.

Graphing a function like this has several benefits:

• Clarity: It provides a clear visual representation of her walking pattern.
• Trends: It helps us easily see how changes in time affect the distance.
• Prediction: Can help make predictions about future values, such as how far she will have walked after 3 or 4 hours.
• Comparison: Multi-line graphs can easily compare different scenarios or rates.

Graphing linear functions, with lines that have no twists or turns, makes these interpretations quick and accessible.

Slope-Intercept Form

The slope-intercept form is a fundamental concept in linear algebra, expressed as \( y = mx + b \). This form makes understanding linear functions simple because it reveals two key elements: the slope \( m \) and the y-intercept \( b \).

In Bianca's walking equation \( d(t) = 3t \), we have:

• Slope \( m = 3 \): This is the rate at which the distance changes per unit of time. It indicates that for every hour Bianca walks, her distance increases by 3 miles. The slope is like the steepness of the line on the graph, clearly illustrating the pace of change. A steeper slope would mean a faster rate.
• Y-intercept \( b = 0 \): This indicates the starting point of the function on the graph, i.e., where the line crosses the y-axis. Since \( b = 0 \), it aligns with the fact that Bianca has covered zero distance at time zero.

The beauty of the slope-intercept form lies in its simplicity and usefulness in both creating linear functions and understanding what they signify. It's straightforward to plot this function on a graph or apply it to real-life situations like calculating distance or money over time.