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Math 142 Lecture Notes for Section 5.6 Section 5.6 - 1 Optimization and Modeling Solving Optimization Problems 101: (1) Introduce variables, look for relationships between those variables, and construct a mathematical model of the form ”Maximinize or minimize f (x) on the interval I. (2) Find the critical values of f (x). (3) Use the methods developed in Section 5.5 to find the absolute maximum (or minimum) of f (x) on I. (4) Use the solutions found to answer the question given. Example 5.6.1: Find two positive numbers whose sum is 55 that maximizes their product. Math 142 Lecture Notes for Section 5.6 2 Example 5.6.2: Find the dimensions of a rectangle with area of 350 centimeters that minimizes the perimeter. Math 142 Lecture Notes for Section 5.6 3 Example 5.6.3: A copy center sells copies at a rate of $.20 a sheet and makes 10, 000 copies per day. A market survey shows that for every $.01 reduction in price, 500 more copies will be made. How much should the copy center charge in order to maximize revenue? Math 142 Lecture Notes for Section 5.6 4 Example 5.6.4: If you are given 50, 000 feet of fencing to create a fenced in square area. What is the maximum rectangular area you can enclose with the fencing? Math 142 Lecture Notes for Section 5.6 5 Example 5.6.5: An open-topped box is going to be made out of a square piece of cardboard that measures 20 inches by 32 inches by cutting identical squares out of the corners and then folding up the flaps. Determine the dimensions of the box in order to maximize the volume of the box. Section 5.6 Suggested Homework 1-9(odd),13,15