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3.4: Optimization Problems Activity Objective: Construct a box from a given piece of paper. Directions: 1. Take a card with a given dimension. 2. Cut squares with side length equal to that on your card from the four corners of your paper. 3. Fold the 4 sides up to create an open topped box. 4. Find the volume of your box. 5. Do you think you could find a box with a larger volume? How? What are the dimensions of the box if we cut 4 squares with side length x from each corner? What is the equation for the volume of the box? What side length will give us a maximum volume? volume? What is the maximum This is how we solve optimization problems. Guidelines: 1. Identify all given quantities and all unknowns 2. Write a primary equation for the quantity to be maximized or minimized 3. Reduce the primary equation to one independent variable. This sometimes requires the use of a secondary equation. 4. Determine the feasible domain. 5. Determine the desired maximum or minimum value by using the calculus techniques we just learned. Ex: The product of two numbers is 72. Minimize the sum of the second number and twice the first number. Ex: Find the points on the graph of y 4 x 2 that are closest to 0, 3. Ex: A new homeowner has 600 meters of fencing to enclose a rectangular portion of the backyard. What should be dimensions of the yard be to maximize the enclosed area? Ex: A solid is formed by adjoining a hemisphere to one end of a right circular cylinder. The total surface area of the solid is 1000 square centimeters. Find the radius of the cylinder that produces the maximum volume.