Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

The cost of sending an overnight package from New York to Atlanta is \(\$ 23.20\) for a package weighing up to but not including 1 pound and \(\$ 2.00\) for each additional pound or portion of a pound. Use the greatest integer function to create a model for the cost C of overnight delivery of a package weighing \(x\) pounds, where \(x>0 .\) Sketch the graph of the function.

Step-by-Step Solution

Answer

The model for the cost of overnight delivery for a package weighing \(x\) pounds is \(C(x) = 23.2 + 2\cdot(\lceil x \rceil - 1)\). This function represents a step-wise increase in cost per each additional pound or part of a pound, starting with an initial cost of $23.20.

Step 1: Interpret the Given Information

According to the problem, the cost of sending a package is $23.20 for the first pound, and then $2.00 for each additional pound or portion of a pound. This means the cost increases by $2.00 each time the weight of the package increases by one pound (or any part of a pound). This increase in cost is represented mathematically by the greatest integer function.

Step 2: Create a Function Model

The function will depend on the weight of the package, \(x\), and will need to account for both the initial cost and the additional cost per pound. Therefore, we can model the cost function as:\[C(x) = 23.2 + 2\cdot(\lceil x \rceil - 1)\]This function represents the initial cost ($23.20) plus the cost per additional pound ($2.00). The greatest integer function, \(\lceil x \rceil\), accounts for each additional pound or part of a pound, and we subtract 1 because the first pound is already included in the initial cost.

Step 3: Sketch the Graph

The graph of the function will be a step function that begins at $23.20 and increases by $2.00 for each additional pound. It will have a horizontal line at each integer value of \(x\) from \(x=1\) onwards, representing the constant cost for each pound. At each integer value of \(x\), there will be a vertical jump of $2.00, representing the additional cost per pound. The graph is not defined for \(x \leq 0\), since the weight of the package must be greater than zero.

Key Concepts

Step Function Graph

Imagine standing on a staircase, and with every step you take, you go up by the same height. Now, visualize this scenario as a graph, and you have the general idea of a step function graph. Specifically, in mathematics, the step function graph represents situations where the value of a function changes suddenly at certain points, creating a series of flat, horizontal 'steps'.

In the context of our shipping cost problem, the cost remains constant at \(23.20 until the weight of the package reaches 1 pound, and then jumps up \)2.00 for every additional pound. Graphically, this appears as a flat line until an increase at every integer value of the weight. These increases are depicted by vertical jumps on the graph, after which the line continues horizontally until the next jump. Unlike continuous functions, where the graph is an unbroken line, the step function graph for our problem has a series of these distinct flat segments and instantaneous jumps, perfectly capturing the discrete nature of shipping costs as the weight of the package increases.

Piecewise Functions

A piecewise function is like a mathematical chameleon, changing its formula depending on the value of the input. It is constructed from multiple sub-functions, each defined on a certain interval. These sub-functions are 'pieces' of the overall function, hence the name 'piecewise'.

Consider our shipping cost example where we have a two-part scenario: a fixed cost up to one pound, and an incremented cost for additional weight. As a piecewise function, the formula for the total cost changes depending on the package weight, which is why we enlist the help of the greatest integer function to model it. In practical terms, piecewise functions are incredibly useful for modelling real-world situations with distinct phases or segments, such as tax brackets, bulk pricing, and yes, even shipping costs. By breaking down complex, real-life situations into simpler parts, piecewise functions make calculation and analysis manageable and more intuitive.

Mathematical Modeling

In essence, mathematical modeling is the craft of turning real-world situations into mathematical terms to analyze, predict, and solve problems. It's like drawing a map that represents the landscape of a particular problem, navigable through the language of mathematics.

For the overnight package delivery from New York to Atlanta, we translate the pricing policy into a mathematical formula, allowing us to predict the cost for any given package weight. The model we use in this case involves the greatest integer function because it captures the essence of charging $2.00 for each additional fraction of a pound. By wrapping the understanding of step functions and piecewise functions into a practical model, we are able to create a tool that can take any weight input and provide the corresponding shipping cost. This model serves as a powerful example of how mathematical concepts can be applied to make sense of and navigate complex, real-world scenarios in an effective and efficient manner.