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Guidelines for Solving Applied Minimum and Maximum Problems 1.Identify all given quantities and quantities to be determined. If possible, make a sketch. 2.Write a primary equation for the quantity that is to be maximized or minimized. 3.Reduce the primary equation to one having a single independent variable. This may involve the use of the secondary equation relating the independent variables of the primary equation.

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Guidelines continued 4.Determine the feasible domain of the primary equation. That is, determine the values for which the stated problem makes sense. 5. Determine the desired maximum or minimum value by calculus techniques, which includes checking the endpoints.

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Example A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to be 1.5 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?

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Example 2 Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are 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?

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Example 3 An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting out a square from each corner then bending up the sides. Find the size of the corner square which will produce a box having the largest possible volume.

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Example 4 A man in a boat at a location 5 miles from the nearest point on a straight beach wishes to reach a place 5 miles along the shore from that point in the shortest possible time. If he can row at a rate of 4 miles an hour and run 6 miles an hour, where should he land?

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Sec 4.7: Optimization Problems EXAMPLE 1 An open-top box is to be made by cutting small congruent squares from the corners of a 12-in.-by-12-in. sheet.

Sec 4.7: Optimization Problems EXAMPLE 1 An open-top box is to be made by cutting small congruent squares from the corners of a 12-in.-by-12-in. sheet.

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