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Warm-Up 1) You have 100 feet of fence and want to make a rectangular pen Ms. Cojohn’s dog Tobin. What are the dimensions of the pen that give Tobin the most room to play? 2) Find two positive numbers whose product is 12 and whose sum is a minimum. Students will be able to solve applied minimum and maximum problems. One of the most common applications of calculus involves determining the maximum or minimum values of a function. While we may be able to easily find a max or min using calculus, setting up the problem might be more difficult. Here are some steps to help – 1) Carefully read the problem and identify given quantities and quantities to be determined. 2) Make a sketch & label it 3) Write down all equations related to your problem/diagram. a. One equation is a primary equation b. One equation is the constraint equation 4) Use the constraint equation to solve for one of the variables. 5) Substitute this equation into the primary equation. 6) Make sure your primary equation only has ONE variable, then differentiate 7) Set the derivative equal to zero & solve! 8) Check your work with the 1st or 2nd Derivative Test Example 1: 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? Example 2: A steel drum in the shape of a right circular cylinder is required to have a volume of 100 ft3. A manufacturer is required to make the drum with the least amount of metal. What dimensions produce a drum with the minimum material? Example 3: A rectangle has one corner on the graph of y 16 x 2 , another at the origin, a third on the positive y-axis, and the fourth on the positive x-axis. For what value of x is the area the largest? 2 Example 4: Which points on the graph of y 4 x are the closest to the point (0,2)? Name: OPTIMIZATION PROBLEMS 1) Joe needs to enclose a rectangular field next to a building with a fence. He has 500 feet of fencing material. One side of the enclosed area is a building and won’t need any fencing. Determine the dimensions of the field that will enclose the largest area. 2) You want to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost $10/ft2 and the material used to build the sides cost $6/ft2. If the box must have a volume of 50ft3. Determine the dimensions that will minimize the cost to build the box. 3) A manufacturer needs to make a cylindrical can that will hold 1.5 liters (equal to 1500 cubic centimeters) of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction. 4) We want to construct a box with a square base and we only have 10 m2 of material to use in construction of the box. Assuming that all the material is used in the construction process determine the maximum volume that the box can have. 5) A printer need to make a poster that will have a total area of 200 in2 and will have 1 inch margins on the sides, a 2 inch margin on the top and a 1.5 inch margin on the bottom. What dimensions will give the largest printed area? 6) We have a piece of cardboard that is 14 inches by 10 inches and we’re going to cut out the corners as shown below and fold up the sides to form a box. Determine the height of the box that will give a maximum volume. Objective: Students will be able to solve applied minimum and maximum problems. Example 1: Jose has 5000 feet of fence. He would like to make a rectangular pen for his dog to play in his backyard. He would like his pen to have the most space for his dog to play. What is the maximum area that can be obtained with 5000 feet of fence? Example 2: A rectangular page is to contain 24 in2 of print. The margins at the top and bottom of the page are to be 1½ inches and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used? Example 3: Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least amount of wire?