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Maximizing Area with Limited Fence: Classic Optimization Problem

In this classic optimization problem, we seek to determine the maximum area that can be enclosed by a given length of fencing. With 40 feet of fence available to surround a rectangular garden adjacent to a barn, we must express the area in terms of a single variable, calculate the first derivative, and identify potential maximum or minimum values. Additionally, we will explore similar examples where optimizing dimensions minimizes surface area or maximizes space, reenforcing the principles of optimization in geometry.

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Maximizing Area with Limited Fence: Classic Optimization Problem

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  1. Optimization 4.7

  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. 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

  6. 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

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

  8. A rectangular field, bounded on one side by a building, is to be fenced in on the other 3 sides. If 3,000 feet of fence is to be used, find the dimensions of the largest field that can be fenced in. w w 3000-2w CV at w = 750 Max at: w = 750 and l = 1500 Max Area = 1,125,000 sq. ft. Always concave down

  9. A physical fitness room consists of a rectangular region with a semicircle on each end. If the perimeter of the room is to be a 200 meter running track, find the dimensions that will make the area of the rectangular region as large as possible. x 2r We want to maximize the area of the rectangle. 2 variables, so lets solve the above equation for r Concave down

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