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3.7 Optimization Problems Solve real-life optimization problems Standard 4.5a A family plans to fence in a rectangular patio area behind their house. They have 120 feet of fence to use. What would be the dimensions of the rectangular region if they wanted to make the patio area enclosed as large as possible? y x Dimensions 30 ft. x 30 ft. A strip of metal 20 inches wide is going to be bent to form an open gutter of rectangular cross section. What dimensions of the cross section will make the carrying capacity maximum? x y when x = 5, y = 20 – 2(5) = 10 dimensions 5 in. x 10 in. A one foot square of metal is to be made into a box without a top by cutting equal squares out of the corners and turning up the edges. What dimensions will make the volume maximum? What is the maximum volume? x x 1 – 2x x x 1 – 2x 1 – 2x x 1 – 2x x x 1 – 2x 1 – 2x x x maximum dimensions: 1/6 x 2/3 x 2/3 minimum volume: 2/27 Express 10 as the sum of two numbers whose product is maximum. x = 5, y = 5 Express 9 as the sum of two positive numbers such that the product of one by the square of the other is a maximum. Minimum Maximum x = 6, y = 3 A page is to contain 30 square inches of print. The margins at the top and bottom of the page are each 2 inches wide. The margins on each side are 1 inch wide. What dimensions will minimize the amount of paper used? 2 1 y x 2 1 Minimum Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 337.5 square centimeters. S 2 x 4 xy 337 .5 2 y y x x 337 .5 2 x 2 4x 2 2 337 .5 2 x 2 V x y x 4 x V 84 .375 x 1 2 x 3 V 84 .375 x dV 84 .375 dx 84 .375 3 2 1 2 3 2 x 3 x 2 x 0 x 56 .25 2 2 x 7.5 2 d V dx 2 3 x 0 for x 7.5 The maximum value occurs when x = 7.5 cm and y = 7.5 cm.