1. Optimization 4.7

2. 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.

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. 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.

5. 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. 

6. 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 :

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. ww 3000-2w CV at w = 750 Always concave down Max at: w = 750 and l = 1500 Max Area = 1,125,000 sq. ft.

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