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3.7 Optimization

Photo by Vickie Kelly, 1999. Greg Kelly, Hanford High School, Richland, Washington. 3.7 Optimization. Buffalo Bill’s Ranch, North Platte, Nebraska. A Classic Problem.

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3.7 Optimization

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  1. Photo by Vickie Kelly, 1999 Greg Kelly, Hanford High School, Richland, Washington 3.7 Optimization Buffalo Bill’s Ranch, North Platte, Nebraska

  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? What’s happening at x = 10? Rel. Maximum!

  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. One application of finding relative extrema is to optimize a function. That is, to find the maximum or minimum values in a particular situation. Key words that might mean optimization: -greatest profit -least cost -least time -greatest strength -least size -optimum size -greatest distance etc…

  5. Make a sketch, if helpful. • Write down all the GIVEN quantities. • Write down quantities that you want to find. • Write down the primary equation for the quantity • that is to be Maximized or Minimized. • Re-write the primary equation as a single variable, • using a secondary equation. • Determine the Feasible Domain. (realistic values) • Find the Max or Min by using Calculus. To find the maximum (or minimum) value of a function:

  6. Example 1: What dimensions for a one liter cylindrical can will use the least amount of material? (Assume can is closed) Motor Oil We can minimize the material by minimizing its surface area. the only given: volume = 1 liter We need secondary equation that relates r and h: Primary Eqtn: (Substitute h into the primary eqtn.)

  7. Example 1: What dimensions for a one liter cylindrical can will use the least amount of material?

  8. Important Points: If the function that you want to optimize has more thanone variable, use substitution (with the secondary eqtn) to rewrite the function to be optimized in terms of one variable. If you are not sure that the extreme you’ve found is a maximum or a minimum, you have to check it. (Use FDT.) If the end points could be the maximum or minimum, you have to check them as well.

  9. Example 2: Which points on the graph of are closest to the point (0, 2)? Use the distance formula: Since d is smallest when the expression under the radical is smallest, we only need to minimize Possible critical numbers:

  10. Using the FDT or SDT, you will see that x = 0 gives a rel. max and both give rel. minimums. So the closest points on the curve to (0,2) are:

  11. Ex. 3 We want to construct a box whose base length is 3 times its base width. The material used to construct the top and bottom of the box costs $10 per square foot. The material used to build the sides of the box costs $6 per sq. ft. If the box must have a volume of 50 ft3 find the dimensions that will minimize the cost of the box. h w l = 3w Primary Eqtn to Minimize: Secondary Eqtn:

  12. Solve for a variable in the secondary to substitute into the primary: Critical #’s would be at: Use SDT to verify there is a minimum at w=1.8821:

  13. So the cost is minimized when width is approx. 1.8821. What would this minimum cost be?

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