Problem 47 - Intermediate Algebra

Question

An online grocery store charges for delivery based on the equation \(C=0.30 p\), where \(C\) represents the cost of delivery in dollars and \(p\) represents the weight of the groceries in pounds. Label the horizontal axis \(p\) and the vertical axis \(C\), and graph the equation \(C=0.30 p\) for nonnegative values of \(p\).

Step-by-Step Solution

Answer

Graph the line using the equation \(C=0.30p\) with point (0,0) and (10,3) for nonnegative values of \(p\).

Step 1: Understanding the Equation

The given equation is \(C = 0.30p\), where \(C\) is the cost and \(p\) is the weight. This tells us that the delivery cost is 30% of the weight of the groceries.

Step 2: Identify the Relationship Type

The equation \(C = 0.30p\) is a linear relationship between the cost \(C\) and the weight \(p\). This means that as the weight \(p\) increases, the cost \(C\) increases proportionally.

Step 3: Determine the Intercepts

To graph the equation, determine the intercepts:- When \(p = 0\), \(C = 0.30 \times 0 = 0\). So, the cost is \(0\) when there is no weight.- There is no vertical intercept as this relationship does not start at any value other than zero.

Step 4: Graph the Line

To graph the line, label the horizontal axis as \(p\) (for weight) and the vertical axis as \(C\) (for cost). Plot the point (0,0) first. Then choose another point, for example when \(p = 10\), \(C = 0.30 \times 10 = 3\). Plot this point (10, 3) on the graph. Draw a straight line through these two points representing the equation.

Step 5: Draw and Label the Graph

Using these points, draw a straight line extending through them for nonnegative values of \(p\). Make sure that \(p\) and \(C\) are labeled on their respective axes. This visually represents the cost's linear increase with weight.

Key Concepts

Linear Relationships

A linear relationship is a fundamental concept in algebra where two variables have a direct proportional connection. In the equation \( C = 0.30p \), the relationship is linear because the cost \( C \) changes in direct proportion to the weight \( p \). This means that as one variable increases, the other does so at a constant rate.
Key characteristics of linear relationships include:

• A constant rate of change, represented by the slope.
• Graphically appears as a straight line.
• For every unit increase in \( p \), \( C \) increases by 0.30 units.

Understanding linear relationships is crucial when analyzing scenarios where one quantity directly affects another. In this case, it helps us predict how the delivery cost varies with the weight of groceries.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most efficient ways to express linear relationships, commonly written as \( y = mx + b \). In this setup, \( m \) is the slope of the line, and \( b \) is the y-intercept.
In the equation \( C = 0.30p \):

• The slope \( m \) is 0.30, indicating that the cost increases by 30 cents for every additional pound.
• The y-intercept \( b \) is 0, which means the graph passes through the origin, reflecting that no cost is incurred with zero weight.

This form emphasizes the linear relationship's simplicity, providing an easy method to quickly determine and visualize how both variables relate. Recognizing the slope-intercept form aids in efficiently sketching graphs of linear equations.

Graphing Techniques

Graphing linear equations involves several straightforward steps that enable us to visualize the relationship between variables like \( C \) and \( p \). For the equation \( C = 0.30p \), the process starts by identifying key points which include the intercepts. These points are critical for a complete and accurate graph.
Steps to graph:

• Label the axes: Horizontal axis \( p \); vertical axis \( C \).
• Identify intercepts: Since \( C(0) = 0 \), the graph starts at the origin (0,0).
• Select additional points: Substitute values of \( p \) to calculate \( C \), such as (10, 3).
• Draw the graph: Connect these points with a straight line, reflecting the linear increase for non-negative \( p \).

By following these techniques, the visual representation of an equation offers insight into how the cost scales with weight, reinforcing comprehension through a graphical approach. Graphing remains a powerful tool to clarify complex relationships.