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3-7: Optimization Problems Objective: 1. To solve applied maximum and minimum problems Assignment: • P. 223-227: 1, 9, 11, 13, 18, 19, 21, 23, 24, 27, 44, 56, 57 You will be able to solve applied maximum and minimum problems The Bigger Box As employees of Lee J. Monroe, we have been tasked with constructing a box with maximum volume. The Bigger Box To make said box, we will take a sturdy sheet of cardstock, which measures 8.5 inches by 11 inches, and cut a square from each corner, keeping in mind that the box must be as voluminous as possible, and then make a box from the resulting net. Optimization Algorithm Identify all given quantities and quantities to be determined, and sketch a picture. Step 1 Write a primary equation for the quantity to be optimized. Step 3 Step 2 Reduce the primary equation to one having a single independent variable using secondary equations. Optimization Algorithm Step 4 Determine a feasible domain for the primary equation. Closed Interval Test Use calculus to determine the minimum or maximum value. First Derivative Test Step 5 Second Derivative Test Exercise 1 A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume? Exercise 2 Which points on the graph of 𝑦 = 4 − 𝑥 2 are closest to the point (0,2)? Exercise 3 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 ½ 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? Exercise 4 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 for ground level to the top of each post. Where should the stake be placed to use the least amount of wire? Exercise 5 Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area? 3-7: Optimization Problems Objective: 1. To solve applied maximum and minimum problems Assignment: • P. 223-227: 1, 9, 11, 13, 18, 19, 21, 23, 24, 27, 44, 56, 57