Optimization Buffalo Bill’s Ranch, North Platte, Nebraska

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Presentation transcript:

Optimization Buffalo Bill’s Ranch, North Platte, Nebraska Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1999

Optimization Practical Problems such as: What’s the most profit you can make? What’s the least amount of materials needed? What’s the shortest distance?

Steps in Solving Optimization Problems Understand the Problem….Read carefully! What is the unknown? What are the given quantities? What are the given conditions? Draw a Picture Identify the givens and the unknowns on the diagram

Steps in Solving Optimization Problems Introduce Notation It may help to use letters that are suggestive of the quantities they represent A for area, h for height, t for time, etc. Use derivatives for rates of change

Steps in Solving Optimization Problems Express what you want to optimize in terms of the other symbols You may have to find relationships (equations) among these variables Use these equations to eliminate all but one variable in the optimization equation

Steps in Solving Optimization Problems Find the absolute maximum or minimum value of the optimization function If the function is on a closed interval, remember to check the endpoints!

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose? There must be a local maximum here, since the endpoints are minimums.

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

We can minimize the material by minimizing the area. Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? Motor Oil We can minimize the material by minimizing the area. We need another equation that relates r and h: area of ends lateral area

Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area

Notes: If the function that you want to optimize has more than one variable, use substitution to rewrite the function. If you are not sure that the extreme you’ve found is a maximum or a minimum, you have to check. If the end points could be the maximum or minimum, you have to check. p