Section 4.4: Modeling and Optimization Objective: Write and solve optimization problems

Critical Point Theorem Suppose a function f is continuous on an interval I and f has exactly one critical number in the interval I, located at x = c. If f has a relative maximum at x = c, then this relative maximum is the absolute maximum of f on the interval I. If f has a relative minimum at x = c, then this relative minimum is the absolute minimum of f on the interval I.

Guidelines for Solving Optimization Problems 1. Draw a diagram, if possible, and label the various unknown quantities that you will need. 2. Write down the information given that is true regardless of the situation. Usually it is a fixed value or some sort of constraint. Write an equation to model this info. 3. Write down a formula for what you are looking to maximize or minimize. 4. Rewrite your equation from step 2 in terms of a single variable. Substitute this expression into what needs to be maximized or minimized. 5. Be sure to check to see if there are any restrictions on the domain. These restrictions could indicate a closed interval. 6. Find the extreme values of the function. Be sure to answer the question being asked.

Example We need to enclose a rectangular field with a fence, and we have 500 ft of fencing. Determine the dimensions of the field that will enclose the largest area.

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?

Alice Gardener wants to build a rectangular enclosure with a dividing fence in the middle of the rectangle. On one side, she plans to put some goats; on the other side she wants to grow some vegetables. The fence along the outside of the rectangle costs $3 per foot, but the dividing fence costs $12 per foot. a.)Alice decides to spend $240 on the fencing, what is the maximum area she can enclose? Justify your answer.

b) If Alice decides that she wants to enclose 300 square feet, what is the minimum cost?

A factory is located 100m downstream and on the opposite side of a river from a power plant. The river is 60m wide. A power line is to be run from the power plant, diagonally down the river and then along the bank to the factory. It costs $130/m to lay the line under the river and $50/m along the bank. Find the point where the line emerges from the river so that the total cost is a minimum.