Chapter 4 | Applications of Derivatives
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4.7 | Applied Optimization Problems Learning Objectives 4.7.1 Set up and solve optimization problems in several applied fields.
One common application of calculus is calculating the minimum or maximum value of a function. For example, companies often want to minimize production costs or maximize revenue. In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume. In this section, we show how to set up these types of minimization and maximization problems and solve them by using the tools developed in this chapter. Solving Optimization Problems over a Closed, Bounded Interval The basic idea of the optimization problems that follow is the same. We have a particular quantity that we are interested in maximizing or minimizing. However, we also have some auxiliary condition that needs to be satisfied. For example, in Example 4.32 , we are interested in maximizing the area of a rectangular garden. Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger. However, what if we have some restriction on how much fencing we can use for the perimeter? In this case, we cannot make the garden as large as we like. Let’s look at how we can maximize the area of a rectangle subject to some constraint on the perimeter. Example 4.32 Maximizing the Area of a Garden A rectangular garden is to be constructed using a rock wall as one side of the garden and wire fencing for the other three sides ( Figure 4.62 ). Given 100 ft of wire fencing, determine the dimensions that would create a garden of maximum area. What is the maximum area?
Figure 4.62 We want to determine the measurements x and y that will create a garden with a maximum area using 100 ft of fencing.
Solution Let x denote the length of the side of the garden perpendicular to the rock wall and y denote the length of the side parallel to the rock wall. Then the area of the garden is A = x · y .
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