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 of tin and bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible? 1

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 of tin and bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible? http://www.math.washington.edu/~conroy/general/boxanim/boxAnimation.htm 2

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 of tin and bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible? 4. Write an equation for the unknown quantity (maximize or minimize). If you can, express the unknown as a function of a single variable or in two equations in two unknowns. This may require considerable manipulation. 5. Find the domain of the single variable function. The possible value of x in the problem 6. Test the critical points and endpoints in the domain of the unknown. Use your knowledge from section 4.1 -4.4 to find the global maximum or global minimum Solving Applied Optimization Problems 1. Read the problem. Read the problem until you understand it. What is given? What is the unknown quantity to be optimized? 2. Draw a picture. Label any part that may be important to the problem. 3. Introduce variables. List every relation in the picture and in the problem as an equation or algebraic expression, and identify the unknown variable. Criticals are x=2, 6 The cutout squares should be 2 in. on a side. 3

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Sec 4.7: Optimization Problems EXAMPLE 1 A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? Solving Applied Optimization Problems 1. Read the problem. 2. Draw a picture 3. Introduce variables. 4. Write an equation for the unknown quantity (maximize or minimize). . 5. Find the domain of the single variable function. 6. Test the critical points and endpoints in the domain of the unknown. Criticals are x= 600 The dimensions is 600X1200 7

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