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4.4 Optimization Buffalo Bill’s Ranch, North Platte, Nebraska Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1999
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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.
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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?
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To find the maximum (or minimum) value of a function: 1 Write it in terms of one variable. 2 Find the first derivative and set it equal to zero. 3 Check the end points if necessary.
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Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? We can minimize the material by minimizing the area. area of ends lateral area We need another equation that relates r and h :
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Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area
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If the end points could be the maximum or minimum, you have to check. 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.
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At the end of his course on mathematical methods in optimization, the professor sternly looks at his students and says: "There is one final piece of advice I'm going to give you now: Whatever you have learned in my course - never ever try to apply it to your personal lives!" "Why?" the students ask. "Well, some years ago, I observed my wife preparing breakfast, and I noticed that she wasted a lot of time walking back and forth in the kitchen. So, I went to work, optimized the whole procedure, and told my wife about it." "And what happened?!" "Before I applied my expert knowledge, my wife needed about half an hour to prepare breakfast for the two of us. And now, it takes me less than fifteen minutes..."
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