14 Lesson Optimization Now youll work some problems where the objective is to optimize a function. That means you want to make it as large as possible or as small as possible depending on the problem. In many problems, youll state the domain before you work the problem. Once you have the function and its domain, youll find the critical points and see if the critical point(s) fall within the domain of the function. You can also use the second derivative test to verify that you have an absolute max or an absolute min in many problems. Example 1: A company that produces digital cameras wants to minimize its production costs. They estimate that their total monthly cost for producing the camera is given by C( x ) = 0.0025x 2 + 80x + 10000 . Find the average cost function. Find the level of
production that results in the smallest average production cost. Use the second derivative test to verify that you have found a minimum cost.
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If the function is not given, the first task is to write a function that describes the situation in the problem. Here are some suggestions to help make this easier: 1. Read the problem carefully to determine what function you are trying to find. 2. If possible, draw a picture of the situation. Choose variables for the values discussed and put them on your picture. 3. Determine if there are any formulas you need to use, such as area or volume formulas. If you have a right triangle in your picture, decide if the Pythagorean Theorem will help. Example 2: A homeowner wants to fence in a rectangular vegetable garden using the back of her garage (which measures 20 feet across) as part of one side of the garden. She has 110 feet of fencing material and wants to use that to build the fence. What should be the dimensions of the garden if she fences in the maximum possible area?
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Example 3: Suppose you wish to fence in a pasture that lies along the straight edge of a river. You will divide the pasture into two parts by means of a fence that runs perpendicular to the river and parallel two of the sides of the pasture. You have 1500 meters of fencing to use, and you wish to fence in the maximum possible area. Determine the dimensions of the pasture that will provide the maximum area. What is that area?
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Example 4: If you cut away equal squares from all four corners of a piece of cardboard and fold up the sides, you will make a box with no top. Suppose you start with a piece of cardboard the measures 3 feet by 8 feet. Find the dimensions of the box that will give a maximum volume.
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Example 5: Postal regulations state that the girth plus length of a package must be no more than 104 inches if it is to be mailed through the US Postal Service. You are assigned to design a package with a square base that will contain the maximum volume that can be shipped under these requirements. What should be the dimensions of the package? (Note: girth of a package is the perimeter of its base.)
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Example 6: You are assigned to design some shipping materials at minimum cost. The package will be a closed rectangular box with a square base, and must have a volume of 50 cubic inches. The material used for the top costs 35 cents per square inch, the material used for the bottom of the box costs 45 cents per square inch, and the material used for the sides costs 20 cents per square inch. Find the surface are of the box and then find the dimensions of the box that will minimize the cost.
From this section, you should be able to Write a function from a verbal description Optimize a function
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