Section 4.5Optimization and Modeling

Steps in Solving Optimization Problems

1. Understand the problem: The first step is to read the problem carefully until it is clearly understood. Ask yourself: What is unknown? What are the given quantities? What are the given conditions?

Steps in Solving Optimization Problems

1. Understand the problem

2. Draw a diagram. In most problems it is useful to draw a diagram and identify the given and required quantities on the diagram. (It might also help to try special cases)

Steps in Solving Optimization Problems

1. Understand the problem

2. Draw a diagram.

3. Introduce Notation. Assign a symbol to the quantity that is to be maximized or minimized (let’s call it Q for now). Also select symbols (a, b, c,…, x, y) for other unknown quantities and label the diagram with these symbols.

Steps in Solving Optimization Problems

1. Understand the problem

2. Draw a diagram.

3. Introduce Notation.

4. Write a formula. Express Q in terms of some of the other symbols.

Steps in Solving Optimization Problems

1. Understand the problem

2. Draw a diagram.

3. Introduce Notation.

4. Write a formula.

5. Write Q as a function of one variable. Use the given information to find relationships (in the form of equations) among the variables. Use these equations to eliminate all but one of the variables.

Steps in Solving Optimization Problems

1. Understand the problem

2. Draw a diagram.

3. Introduce Notation.

4. Write a formula.

5. Write Q as a function of one variable.

6. Find the domain of this function

Steps in Solving Optimization Problems

1. Understand the problem

2. Draw a diagram.

3. Introduce Notation.

4. Write a formula.

5. Write Q as a function of one variable.

6. Find the domain of this function

7. Find the global maximum or global minimum value of this function.

Example

• You have to enclose 3200 square feet of land (in a rectangle) using a fence on 3 sides that will cost $2 per foot, and a special wall on the fourth side that will cost $6 per foot. What are the dimensions of your enclosure that will minimize your cost?

Example

• A box with a square base and an open top needs to be constructed that will have a volume of 3200m3. What should be the dimensions of the box so that the surface area will be minimized?– Hint: draw a picture of the situation

• A rectangle is to be inscribed within a right triangle with a base of 3 and a height of 4. What is the largest rectangle that can be created?

Example

Example

• A cylinder is to be created within a sphere of radius r. What should are the dimensions of the cylinder which maximize its volume?