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**3.7 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? 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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**Procedure for Solving Optimization Problems**

Assign symbols to all given quantities and quantities to be determined. Write a primary equation for the quantity to be maximized or minimized. Reduce the primary equation to one having ONE independent variable. This may involve the use of secondary equation. Determine the domain. Make sure it makes sense. Determine the max or min by differentiation. (Find the first derivative and set it equal to zero) If the maximum or minimum could be at the endpoints, you have to check them.

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**We can minimize the material by minimizing the area.**

Example: 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

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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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**A manufacturer wants to design an open box **

having a square base and a surface area of 108 in2. What dimensions will produce a box with maximum volume? Since the box has a square base, its volume is V = x2h h Note: We call this the primary equation because it gives a formula for the quantity we wish to optimize. x x The surface area = the area of the base + the area of the 4 sides. S.A. = x2 + 4xh = 108 We want to maximize the volume, so express it as a function of just one variable. To do this, solve x2 + 4xh = 108 for h.

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**3x2 = 108 Substitute this into the Volume equation.**

To maximize V we find the derivative and it’s C.N.’s. 3x2 = 108 We can conclude that V is a maximum when x = 6 and the dimensions of the box are 6 in. x 6 in. x 3 in.

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**Find the points on the graph of y = 4 – x2 that are closest**

to the point (0,2). What is the distance from the point (x,y) and (0,2)? 4 3 2 1 (x,y) (x,y) We want this dist. to be minimized.

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**= 2x(2x2 – 3) = 0 C.N.’s imaginary dec. dec. inc. inc. 1st der. test**

dec. dec. inc. inc. 1st der. test minimum minimum Two closest points.

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