1. Modeling and Optimization Chapter 5.4

2. 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

3. Example 1: Using the Strategy Find two numbers whose sum is 20 and whose product is as large as possible. 3

4. Example 1: Using the Strategy 4

5. 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

6. Example 2: Inscribing Rectangles 6

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9. Example 3: Fabricating a Box 9

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11. 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

12. Example 4: Designing a Can 12

13. Example 4: Designing a Can 13

14. Example 4: Designing a Can 14

15. Example 4: Designing a Can 15

16. Example 4: Designing a Can 16

17. Examples from Economics 17

18. Examples from Economics 18

19. Example 5: Maximizing Profit 19

20. Example 5: Maximizing Profit 20

21. Example 5: Maximizing Profit 21

22. Example 5: Maximizing Profit 22

23. Minimizing Average Cost 23

24. Minimizing Average Cost 24

25. Example 6: Minimizing Average Cost 25

26. Example 6: Minimizing Average Cost 26

27. Example 6: Minimizing Average Cost 27

28. 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

29. Example 7: Minimum Time 29

30. Example 7: Minimum Time 30

31. Example 7: Minimum Time 31

32. Example 7: Minimum Time 32

33. Example 7: Minimum Time 33

34. Exercise 5.4 34