Applied Max and Min Problems (Optimization) 5.5. Procedures for Solving Applied Max and Min Problems 1.Draw and Label a Picture 2.Find a formula for the.

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Presentation transcript:

Applied Max and Min Problems (Optimization) 5.5

Procedures for Solving Applied Max and Min Problems 1.Draw and Label a Picture 2.Find a formula for the quantity that is to be maximized or minimized 3.Eliminate variables by expressing the quantity to be maximized or minimized as a function of one variable 4.Find the interval of possible values based upon physical restrictions (Domain) 5.Use techniques from 5.4 to find the max or min

Example 1 (from pg. 310) A garden is to be laid out in a rectangular area and protected by a chicken wire fence. What is the largest possible area of the garden if only 100 running feet of chicken wire is available for the fence?

Try this one A garden is to be laid out in a rectangular area and protected by a fence. What is the largest possible area of the garden if only 60 feet of fence is available?

Example 2 (from pg. 311) An open box is to be made from a 16 inch by 30 inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain the box with the largest volume? What is the greatest volume that can be obtained?

Try This… An open box is to be made from a 10 inch by 12 inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain the maximum volume? Hint: You will need to use your calculator to estimate the root from the quadratic formula Answer: x = 1.81 Max Volume = 37.6 cu. inches

Difficult Example An offshore oil well is located at a point W that is 5km from the closest point A on a straight shoreline. Oil is to be piped from W to a shore point B that is 8km from A by piping it on a straight line under water from W to some shore point P between A & B and then onto B via pipe along the shoreline. If the cost of laying pipe is $1,000,000/km under water and $500,000/km over land, where should the point P be located to minimize the cost of laying the pipe?

Practice Pg. 318 and 319 (3, 5, 11, 19)