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# Calc04_4 - 4.4 Modeling and Optimization Buffalo Bills...

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4.4 Modeling and Optimization Buffalo Bill’s Ranch, North Platte, Nebraska Greg Kelly, Hanford High School, Richland, Washington Photo 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? x x 40 2 x - ( 29 40 2 A x x = - 2 40 2 A x x = - 40 4 A x = - 0 40 4 x = - 4 40 x = 10 x = 40 2 l x = - w x = 10 ft w = 20 ft l = 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? x x 40 2 x - ( 29 40 2 A x x = - 2 40 2 A x x = - 40 4 A x = - 0 40 4 x = - 4 40 x = 10 x = ( 29 10 40 2 10 A = - ( 29 10 20 A = 2 200 ft A = 40 2 l x = - w x = 10 ft w = 20 ft l =

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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.
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. 2 2 2 A r rh π π = + area of ends lateral area

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Calc04_4 - 4.4 Modeling and Optimization Buffalo Bills...

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