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OPTIMIZATION PROBLEMS (Area and Volume) optimize: to realize the best possible outcome subject to a set of restrictions (the domain of the function is usually restricted) An optimization problem can be solved using a mathematical model that is developed using information given in the problem. The numerical solution represents the extreme value (max or min) of the model. Algorithm for Solving Optimization Problems: 1. Understand the problem and whenever possible, draw a diagram, labelling the given and required quantities. 2. Determine a function in one variable that represents the quantity to be optimized. 3. Determine the domain of the function to be optimized. 4. Use the algorithm for extreme values to determine the absolute maximum or minimum value in the domain. 5. Use the result from step #4 to solve the problem. Example BARN A farmer has 800 m of fencing and wishes to enclose a rectangular field with one side along the side of the barn. Determine the dimensions of the field of maximum area. Example A rectangular open topped box is to be made from a piece of cardboard measuring 18 cm by 48 cm by cutting a square from each corner and turning up the sides. Determine the size of the squares that must be removed from the cardboard in order to maximize the volume of the box. State the resulting dimensions of the box. Example A pop can in the shape of a cylinder is to have a capacity of 250 cm3. The diameter of the can must be no less than 4 cm and no greater than 8 cm. Determine the dimensions that would minimize the amount of material needed for its construction. Vcylinder = r2h SAcylinder = 2r2 + 2rh h d Homework: p.145–147 #1, 3–9, 11a, 19