Example 1

EXAMPLE 1 Making Skateboards and Dolls

A toy manufacturer can manufacture only skateboards, only dolls, or some mixture of skateboards and dolls. Skateboards require 5 units of plastic and can be sold for a profit of $1, while dolls require 2 units of plastic and can be sold for a $0.55 profit. If 60 units of plastic are available, what numbers of skateboards and/or dolls should be manufactured for the company to maximize its profit?

Attacking this and other mixture problems requires carrying out a series of steps that determine the essence of the problem.

As a first step, we need to construct a mathematical model to take the “verbal” information that we have been given and display it in a form that makes it easier to convert into the mathematics necessary to solve the problem. This is done by making a mixture chart for the information we are given (see Figure 4.1).

Figure 4.1: Figure 4.1 Mixture chart for Example 1.

In the rows of this chart, we display the products we want to make, and in the column of the chart, we display the resources and the profit margin information that is available. In this case, we have two products, so we have two rows. We have one resource, which accounts for there being one column.

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The other column is reserved for profit information. Since this information is used in a somewhat different way from the information about the resources, we will separate the resource column(s) from the profit column by a double bar.

Because we do not know the number of skateboards the company should make, we will use a letter to represent the unknown number of skateboard units that the company might manufacture. Similarly, will represent the number of dolls that the company might manufacture. We enter these letters as part of the labels of the rows of our table.

We can now enter the numbers about resources in the columns based on the information we have been given. in this case, there is one resource: plastic. Thus, for the 5 units of plastic needed for a skateboard, we record a 5 in the skateboards row and the containers of plastic column. Similarly, we enter a 2 in the second row and first column because dolls require 2 units of plastic each. Because we have 60 units of plastic available, we display this fact by placing the number 60 at the top of this column. We complete the table with the information about profit. We enter $1 in the skateboards row and profit column and $0.55 in the dolls row and profit column.