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Optimization Techniques for Maximizing Area Enclosed by Fence

Explore classic problem scenarios to optimize area enclosed by a fence using derivatives and critical points. Learn substitution methods and endpoint checks.

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Optimization Techniques for Maximizing Area Enclosed by Fence

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  1. Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College Photo by Vickie Kelly, 1999 Modeling and Optimization Buffalo Bill’s Ranch, North Platte, Nebraska

  2. 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?

  3. 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?

  4. 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. To find the maximum (or minimum) value of a function:

  5. 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

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

  7. 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

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