3.7 Optimization Problems

Objective: Solve applied minimum and maximum problems.

Miss BattagliaAB/BC Calculus

We need to enclose a field with a fence.  We have 500 feet of fencing material and a building is on one side of the field and so won’t need any fencing.  Determine the dimensions of the field that will enclose the largest area.

1. Identify all given quantities and all 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 secondary equations relating the independent variables of the primary equation.

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 the calc techniques we learned this chapter.

Guidelines for Solving Applied Minimum and Maximum Problems

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?

Finding Maximum Volume

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We want to construct a box whose base length is 3 times the base width.  The material used to build the top and bottom cost $10/ft2 and the material used to build the sides cost $6/ft2.  If the box must have a volume of 50ft3 determine the dimensions that will minimize the cost to build the box.

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 from ground level to the top of each post. Where should the stake be placed to use the least amount of wire?

Read 3.7 Page 223 #4, 9, 17, 20, 21, 23, 26, 29, 42, 39

Classwork/Homework