Let’s look at how we can maximize the area of a rectangle subject to some constraint on the perimeter. 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. Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger. For example, in Example 4.32, we are interested in maximizing the area of a rectangular garden. However, we also have some auxiliary condition that needs to be satisfied. We have a particular quantity that we are interested in maximizing or minimizing. The basic idea of the optimization problems that follow is the same. Solving Optimization Problems over a Closed, Bounded Interval 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. In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume. For example, companies often want to minimize production costs or maximize revenue. One common application of calculus is calculating the minimum or maximum value of a function. 4.7.1 Set up and solve optimization problems in several applied fields.
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