Modeling and Optimization Chapter 5.4 Strategy for Solving Min-Max Problems 1.Understand the problem 2.Develop a mathematical model of the problem 3.Graph the function* 4.Identify the critical points and endpoints 5.Solve the mathematical model 6.Interpret the solution 2 Example 1: Using the Strategy Find two numbers whose sum is 20 and whose product is as large as possible. 3 Example 1: Using the Strategy 4 Example 2: Inscribing Rectangles A rectangle is to be inscribed under one arch of the sine curve. What is the largest area the rectangle can have, and what dimensions give that area? 5 Example 2: Inscribing Rectangles 6 7 8 Example 3: Fabricating a Box 9 10 Example 4: Designing a Can You have been asked to design a one-liter can shaped like a right circular cylinder. What dimensions will use the least material? 11 Example 4: Designing a Can 12 Example 4: Designing a Can 13 Example 4: Designing a Can 14 Example 4: Designing a Can 15 Example 4: Designing a Can 16 Examples from Economics 17 Examples from Economics 18 Example 5: Maximizing Profit 19 Example 5: Maximizing Profit 20 Example 5: Maximizing Profit 21 Example 5: Maximizing Profit 22 Minimizing Average Cost 23 Minimizing Average Cost 24 Example 6: Minimizing Average Cost 25 Example 6: Minimizing Average Cost 26 Example 6: Minimizing Average Cost 27 Example 7: Minimum Time A man is in a boat 2 miles from the nearest point on the coast. He is to go to a point Q, located 3 miles down the coast and 1 mile inland. He can row at 2 miles per hour and walk at 4 miles per hour. Toward what point on the coast should he row in order to reach the point Q in the least time? 28 Example 7: Minimum Time 29 Example 7: Minimum Time 30 Example 7: Minimum Time 31 Example 7: Minimum Time 32 Example 7: Minimum Time 33 Exercise