Problem 61 - College Algebra and Calculus: An Applied Approach

Question

A shipping company's charge for sending an overnight package from New York to Atlanta is \(\$ 12.80\) for the first pound and \(\$ 2.50\) for each additional pound or fraction thereof. Use the greatest integer function to create a model for the charge \(C\) for overnight delivery of a package weighing \(x\) pounds. Use a graphing utility to graph the function, and discuss its continuity.

Step-by-Step Solution

Answer

The model for the charge for overnight delivery of a package weighing \(x\) pounds is \(C= \$12.80 + \$2.50 \cdot \lfloor{x}\rfloor\). The graph of this function is a step function and is discontinuous at every integer value of \(x\), and it is right continuous.

Step 1: Determine Base Charge

The base charge to send a package is \(\$ 12.80\). This is the charge for the first pound.

Step 2: Determine Additional Charge

For each additional pound or fraction thereof, an extra charge of \(\$ 2.50\) is to be paid.

Step 3: Formulate the Model

The total cost \(C\) to ship a package of weight \(x\) pounds is given by the equation \(C= \$12.80 + \$2.50 \cdot \lfloor{x}\rfloor\). Note that we use the floor function represented by \lfloor{x}\rfloor, which yields the greatest integer less than or equal to \(x\). This is because, an additional pound or fraction thereof incurs the same additional charge.

Step 4: Discuss Continuity

Plotting the equation, we have a step function which is discontinuous at every integral value of \(x\). The function is right continuous, meaning for each positive integer \(n\), the value of the function approaches \(n\) from the left but jumps to \(n+1\) at \(x=n\).

Key Concepts

Discontinuity

A discontinuity in a function occurs when there is an abrupt change in the value of the function at certain points. In the context of our shipping cost problem, this happens at every whole number of pounds due to the pricing structure. The cost function is defined using a step function, where the price jumps at those specific points.
For example, a package weighing just under 2 pounds incurs the same added charge as one weighing exactly 2 pounds, but the moment the weight hits 2 pounds, the shipping cost increases by another increment of \\(2.50\\). This jump in cost illustrates a discontinuity, making the graph appear in steps. These jumps are visualized as breaks or gaps at integral values when plotted on a graph.
Discontinuous functions like this one make modeling real-world phenomena like shipping or billing services intuitive, yet they pose a challenge when predicting slight changes around these points.

Step Function

A step function is a type of discontinuous function characterized by segments of constant values. In the scenario of the shipping charges, each segment corresponds to weights in integer ranges. For a better understanding, consider how the graph would look for our equation \(C = \\(12.80 + \\)2.50 \cdot \lfloor{x}\rfloor \).

• From \(1\) to just under \(2\) pounds, the cost remains constant.
• At exactly \(2\) pounds, the cost jumps to a new constant value.

This pattern repeats itself at every subsequent integer increase in weight, creating a sequence of flat horizontal line segments on the graph.
Each segment represents the stable cost for that weight range, and the jumps between segments illustrate the additional charge incurred. Understanding step functions provides clarity into how stair-step-like graphs represent real-world scenarios such as billing and sales, reflecting the incremental charges introduced at specific thresholds.

Floor Function

The floor function, denoted as \(\lfloor{x}\rfloor\), is crucial in computing shipping charges here because it helps determine costs based on weight. It works by rounding down to the nearest integer less than or equal to \(x\). This property allows us to incorporate the 'pound or fraction of a pound' rule seamlessly.

For example, whether a package weighs \(3.1\) pounds or exactly \(3\) pounds, the floor function \(\lfloor{3.1}\rfloor\) equals \(3\), making the charge equivalently based on \(3\) pounds.
In our cost equation, \(\\(12.80 + \\)2.50 \cdot \lfloor{x}\rfloor\), the floor function ensures that every additional pound, or any fraction thereof, incurs the appropriate \\(2.50\\). This appropriately models how shipping fees are calculated in practical scenarios like this one.
The floor function is handy in many computational applications, especially when discrete and whole integer results are required from continuous or floating numbers.