Problem 96 - College Algebra

Question

A cargo service charges a flat fee of \(\$ 4\) plus \(\$ 1\) for each pound or fraction of a pound to mail a package. Let \(C(x)\) represent the cost to mail a package that weighs \(x\) pounds. Graph the cost function on the interval \((0,5].\)

Step-by-Step Solution

Answer

The graph of the cost function \(C(x)\) will be a straight line that starts from (0,4) and has a slope of $1. It will go through points such as (1,5), (2,6), (3,7), (4,8), and (5,9). This line shows how the cost to mail a package increases by $1 for each additional pound of weight.

Step 1: Understand the Function

First, let's identify the components of linear function \(C(x)\). This function represents the cost to mail a package that weighs \(x\) pounds. The flat fee (y-intercept) is $4, and the cost per pound (slope) is $1. Therefore, our linear function is \(C(x) = 4 + 1x\).

Step 2: Create a Table of Values

Next, create a table of values to plot on the graph. Since the problem asks to graph the function on the interval \((0,5]\), let's use weights (values of \(x\)) from 1 to 5. Calculate the cost (\(C(x)\)) for each corresponding weight. These will create the points you plot on the graph. Remember, even if the weight is a fraction, the cost should be calculated as if it's a whole pound.

Step 3: Plot the Points on the Graph

Plot the table of values on the graph. The weight (\(x\)-values) will be on the horizontal axis, and the cost (\(C(x)\)) will be on the vertical. Plot each point based on its coordinates from the table of values.

Step 4: Draw the Line and Label the Graph

Draw a line through the points plotted on the graph. Label the axes appropriately (e.g., 'Weight in pounds' and 'Cost in dollars'), and include a title.

Key Concepts

Cost Functions

Cost functions are mathematical expressions used to determine the total cost associated with a particular activity or operation. In our given example, the cost function is represented as a linear function. Linear functions take the form \(C(x) = mx + b\), where \(m\) represents the slope or rate of change, and \(b\) signifies the y-intercept or starting value. For the cargo service scenario, the cost function allows us to input the weight of the package and calculate the cost to mail it.

By identifying the components of the cost function correctly, we can easily calculate costs and make informed decisions. The flat fee of \\(4 is the y-intercept, meaning it is the starting cost regardless of the package's weight. The \\)1 charge per pound represents the slope or rate of change. Together, they form the function \(C(x) = 4 + 1x\). Understanding this function helps in predicting how costs increase with weight, making it a valuable tool for budgeting and financial planning.

Graphing

Graphing linear functions is a key step in visualizing how variables relate to one another. To graph a cost function, start by creating a table of values that represent different points, in this case, weights of packages.

• The x-values are the weights of the packages in pounds.
• The y-values are the total costs to mail the packages.

By doing this, you can plot each point on a coordinate plane, with the x-axis representing the weight and the y-axis representing the cost. For example, if a package weighs 1 pound, the cost is \(\\(5\); for 2 pounds, it's \(\\)6\), and so on, up to 5 pounds. After plotting these points, draw a straight line that connects them.

This line, known as the graph of the function, visually demonstrates the relationship between package weight and cost. By analyzing the graph, one can easily see how costs accumulate, making it simpler to predict costs for any given weight.

Slope and Intercept

In the context of linear functions, the slope and intercept are essential components that define the line's characteristics. The slope, represented as \(m\) in the linear equation \(C(x) = mx + b\), indicates how much the cost changes with each additional unit of weight. For the cargo service example, the slope is \\(1 per pound, meaning the cost increases by \\)1 for every extra pound mailed.

The y-intercept, denoted as \(b\), is the point where the line intersects the y-axis. It represents the starting cost before any additional weights are factored in. In our cost function, the y-intercept is \\(4, which is the flat fee. This means regardless of whether the package has any weight or not, the base charge is \\)4.

• The slope tells us about the rate of change in cost.
• The intercept tells us about the initial fixed cost.

Understanding both these elements help in constructing accurate predictions and assessments of costs.

Real-World Applications

Linear functions, such as the one represented in the cargo service scenario, are not just limited to theoretical exercises; they have practical real-world applications too. Cost functions are widely used in business and economics to model scenarios such as revenue generation, expense tracking, and profit analysis.

For instance, a company might use a similar cost function to estimate shipping costs, enabling them to set competitive yet profitable pricing strategies. By understanding the cost function and its graph, businesses can:

• Predict the financial impact of changes in production or shipping volume.
• Determine pricing strategies that maximize profits while remaining competitive.
• Identify opportunities for cost reduction by analyzing how costs change with different variables.

Overall, mastering the use of linear functions can provide a significant advantage in both personal financial planning and professional strategic decision-making.