Problem 89 - Beginning and Intermediate Algebra

Question

To ship small packages within the United States a shipping company charges \(\$ 3.75\) for the first pound and S1.10 for each additional pound or fraction of a pound. Let \(C(x)\) represent the cost of shipping a package, and let \(x\) represent the weight of the package. Graph \(C(x)\) for any package weighing up to (and including) 6 Ib.

Step-by-Step Solution

Answer

The cost function \(C(x)\) can be defined as a piecewise function: \(C(x) = \begin{cases} 3.75, & 0 \le x \le 1 \\ 3.75 + 1.10(x-1), & 1 < x \le 6 \end{cases} \) Evaluating this function at several points, we get the following values: 1. x = 0: \(C(0) = 3.75\) 2. x = 1: \(C(1) = 3.75\) 3. x = 2: \(C(2) = 4.85\) 4. x = 3: \(C(3) = 5.95\) 5. x = 4: \(C(4) = 7.05\) 6. x = 5: \(C(5) = 8.15\) 7. x = 6: \(C(6) = 9.25\) Plotting these points on a graph with the x-axis representing the package weight and the y-axis representing the shipping cost gives us a visual representation of the cost function for package weights up to 6 pounds.

Step 1: Define the Cost Function

We can formulate the cost function C(x) as a piecewise function: \(C(x) = \begin{cases} 3.75, & 0 \le x \le 1 \\ 3.75 + 1.10(x-1), & 1 < x \le 6 \end{cases} \) where x represents the weight of the package.

Step 2: Evaluate for Different Values of x

To create the graph, let's compute C(x) for various values of x within the range [0, 6]: 1. x = 0: \(C(0) = 3.75\) 2. x = 1: \(C(1) = 3.75\) 3. x = 2: \(C(2) = 3.75 + 1.10(2-1) = 4.85\) 4. x = 3: \(C(3) = 3.75 + 1.10(3-1) = 5.95\) 5. x = 4: \(C(4) = 3.75 + 1.10(4-1) = 7.05\) 6. x = 5: \(C(5) = 3.75 + 1.10(5-1) = 8.15\) 7. x = 6: \(C(6) = 3.75 + 1.10(6-1) = 9.25\) Now we have several points that can be plotted on the graph.

Step 3: Plot the Graph

Using the computed points, plot the cost function C(x) on the x-y plane, where the x-axis represents the weight of the package (x), and the y-axis represents the cost of shipping (C(x)). From the plot, we can observe the relationship between the package weight and the cost for shipping within the United States using the given shipping company's charges. Remember that for packages weighing over 1 pound, the cost will increase by $1.10 for each additional pound or a fraction of a pound.

Key Concepts

Cost Function

A cost function is a mathematical description that explains how costs change with varying factors, like the weight of a package. In the given problem, the cost of shipping a package within the United States is expressed as a piecewise function. A piecewise function is used when different formulas apply to different sections of the domain. Here, the cost function is defined as:

\[ C(x) = \begin{cases} 3.75, & 0 \le x \le 1 \ 3.75 + 1.10(x-1), & 1 < x \le 6 \end{cases} \]

This function shows that for packages weighing up to 1 pound, the cost is a flat rate of \(3.75. When packages weigh more than 1 pound, the cost includes an additional \)1.10 for each additional pound. Such functions help determine shipping costs based on package weight, allowing companies to price their services accurately.

Graphing Functions

Graphing a function allows us to visually interpret the relationship between variables. In this exercise, the graph illustrates how shipping costs change with package weight. To graph the function \( C(x) \), we calculate the cost for various weights \( x \) between 0 and 6 pounds:

• 0 pounds, \( C(0) = 3.75 \)
• 1 pound, \( C(1) = 3.75 \)
• 2 pounds, \( C(2) = 4.85 \)
• 3 pounds, \( C(3) = 5.95 \)
• 4 pounds, \( C(4) = 7.05 \)
• 5 pounds, \( C(5) = 8.15 \)
• 6 pounds, \( C(6) = 9.25 \)

Next, plot these points on a coordinate plane, where the x-axis represents the package weight and the y-axis represents the shipping cost. Connecting these points reveals the step-like graph typical of piecewise functions. It reflects the constant rate up to 1 pound and incremental costs for weights beyond that, making it useful for quick cost estimation.

Weight and Cost Relationship

Understanding the weight and cost relationship is crucial for efficiently managing shipping expenses. In this exercise, the relationship is direct and predictable, as defined by the piecewise cost function. As weight increases beyond the first pound:

• Each full or partial additional pound results in a constant increase of $1.10 in the cost.
• The initial flat rate of $3.75 covers any package weighing 1 pound or less.

It is essential to understand how each pound impacts the total cost. This knowledge aids decision-making in shipping logistics, assisting both consumers and businesses in optimizing their shipping practices. By analyzing the weight-cost relationship graph, one can identify cost-effective strategies based on package weight, leading to smarter shipping choices.