Problem 37 - Algebra for College Students

Question

An online grocery store charges for delivery based on the equation \(C=0.30 p\), where \(C\) represents the cost 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 a straight line from (0,0) through points like (5,1.5) and label axes.

Step 1: Identify Variables and Constants

In the equation \(C = 0.30 p\), \(C\) is the delivery cost in dollars, and \(p\) is the weight in pounds. The number 0.30 is the fixed rate per pound.

Step 2: Set Up the Coordinate Axes

The horizontal axis (x-axis) is labeled as \(p\) and represents the weight of the groceries in pounds. The vertical axis (y-axis) is labeled as \(C\) and represents the cost in dollars.

Step 3: Determine Graph Range

Since we are asked to graph for nonnegative values, consider \(p \geq 0\). Therefore, the graph should start from the origin (0,0) and extend to positive values on both axes.

Step 4: Find and Plot Key Points

Choose a few values of \(p\) to calculate \(C\), e.g., \(p = 0, 5, 10\). Calculate the corresponding \(C\) values: \(C(0) = 0.30 \times 0 = 0\), \(C(5) = 0.30 \times 5 = 1.5\), \(C(10) = 0.30 \times 10 = 3\). Plot the points: (0,0), (5,1.5), and (10,3) on the graph.

Step 5: Draw the Graph Line

Connect the plotted points with a straight line. Since the equation \(C = 0.30 p\) is linear, the graph is a straight line passing through the origin with a slope of 0.30.

Step 6: Label the Graph

Ensure the axes have labels: \(p\) for the horizontal axis and \(C\) for the vertical axis. Mark the scale on both axes for clarity, showing values at consistent intervals.

Key Concepts

Coordinate Axes

In graphing linear equations, understanding the coordinate axes is crucial. The coordinate system is composed of two perpendicular lines: the horizontal line and the vertical line.

• The horizontal line is known as the x-axis.
• The vertical line is referred to as the y-axis.

Both axes are critical for locating points on a plane using ordered pairs, often expressed as \((x, y)\). In the context of our problem, the x-axis is labeled as \(p\), representing the weight of groceries in pounds, and the y-axis is labeled as \(C\), representing the cost in dollars. To properly graph any equation, ensure both axes include labels and scales to provide context for the values plotted.

Slope

The slope of a line is a measure that explains the steepness or incline of the line. It indicates how much the dependent variable (cost, in our case) will change for a change in the independent variable (weight in our case).

• In the equation \(C = 0.30 p\), the slope is 0.30.
• This number represents the cost increase per pound of grocery weight.

The slope can be thought of as "rise over run"—for every unit increase in \(p\) (weight), \(C\) (cost) rises by 0.30 units. Understanding the slope is paramount as it defines the relationship between the variables
and helps in predicting how changes in one variable affect the other.

Linear Function

A linear function, like the one given by \(C = 0.30 p\), is an algebraic equation where each term is either a constant or the product of a constant and a single variable. This type of function forms a straight line when graphed.

• The general form of a linear function is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
• Our equation's format lacks the \(b\) term, making it a line through the origin.

Linear functions are straightforward and predictable, making them ideal for modeling simple relationships, such as the cost increasing with weight. Recognizing the structure of linear equations can help simplify the process of solving and graphing them.

Graph Plotting

Plotting a graph begins by identifying the points that fit the equation. Choose values for the independent variable \(p\), calculate the dependent variable \(C\), and then plot these coordinates on the graph.

• For \(p = 0\), \(C = 0\).
• For \(p = 5\), \(C = 1.5\).
• For \(p = 10\), \(C = 3\).

These points, \((0,0)\), \((5,1.5)\), and \((10,3)\), are marked on the plane.Once the points are plotted, draw a straight line through them.
This line not only visualizes our linear function but also represents the relationship between weight and cost as described by our equation, concluding with labeling the axes and marking units for clarity. This systematic approach ensures accurate representation
of the data and function being analyzed.