Identify the domain of consideration for the function in step \(4\) based on the physical problem to be solved. Write any equations relating the independent variables in the formula from step \(3.\) Use these equations to write the quantity to be maximized or minimized as a function of one variable. This formula may involve more than one variable. Write a formula for the quantity to be maximized or minimized in terms of the variables. If applicable, draw a figure and label all variables.ĭetermine which quantity is to be maximized or minimized, and for what range of values of the other variables (if this can be determined at this time). Problem-Solving Strategy: Solving Optimization Problems. Now let's look at a general strategy for solving optimization problems similar to Example 4.135. Differentiating the function \(A(x),\) we obtain At the endpoints, \(A(x)=0.\) Since the area is positive for all \(x\) in the open interval \((0,50),\) the maximum must occur at a critical point. These extreme values occur either at endpoints or critical points. Maximize \(A(x)=100x-2x^2\) over the interval \(.\)Īs mentioned earlier, since \(A\) is a continuous function on a closed, bounded interval, by the extreme value theorem, it has a maximum and a minimum. Therefore, we consider the following problem: Therefore, let's consider the function \(A(x)=100x-2x^2\) over the closed interval \(.\) If the maximum value occurs at an interior point, then we have found the value \(x\) in the open interval \((0,50)\) that maximizes the area of the garden. However, we do know that a continuous function has an absolute maximum (and absolute minimum) over a closed interval. Therefore, we need \(x\gt 0\) and \(y\gt 0.\) Since \(y=100-2x,\) if \(y\gt 0,\) then \(x\lt 50.\) Therefore, we are trying to determine the maximum value of \(A(x)\) for \(x\) over the open interval \((0,50).\) We do not know that a function necessarily has a maximum value over an open interval. To construct a rectangular garden, we certainly need the lengths of both sides to be positive. What is the maximum area?īefore trying to maximize the area function \(A(x)=100x-2x^2,\) we need to determine the domain under consideration. Given \(100\) ft of wire fencing, determine the dimensions that would create a garden of maximum area. Maximizing the Area of a Garden.Ī 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 4.136). 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 4.135, 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. Subsection 4.7.1 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. Set up and solve optimization problems in several applied fields. Section 4.7 Applied Optimization Problems Learning Objectives.
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