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Class Note for MATH 1314 at UH 2


Class Note for MATH 1314 at UH 2

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 21 views.

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Date Created: 02/06/15
Math 1314 Lesson 14 Optimization Now you ll 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 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 nd 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 In many problems you ll state the domain before you work the problem Once you have the function and its domain you ll nd the critical points and see if the critical points 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 man would like to have a rectangular shaped garden in his back yard He has 100 feet of fencing to use to fence in the garden Find the dimensions for the largest possible garden he can make if he uses all of the fencing Example 2 If you cut away equal squares from all four comers 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 4 feet by 5 feet Find the dimensions of the box that will give a maximum volume Example 3 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 Example 4 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 Example 5 A power station and a factory are on opposite sides of a river 60 m wide A cable must be run from the power station to the factory The factory is 1000 m downstream from a point directly across the river from the power station It costs 25 per meter to run the cable under water and 15 to run the cable on land Determine how far downstream from the power station the construction crew should begin laying the cable under land in order to minimize construction costs 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 dimensions of the box that will minimize the cost Example 7 When an apartment leasing company sets its rent for a onebedroom apartment at 600 per month all of the 50 onebedroom apartments are leased Research shows that for every 25 increase in monthly rent they will lose one tenant What should be the rent if the leasing company wants to maximize its revenue Example 8 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 From this section you should be able to Write a function from a verbal description Optimize a function


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