Name: Block: Date: FOM 11

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Ch. 6 Lesson 9 - Optimization Problems III : Linear Programming

The solutions to an optimization problem are always found at one of _____________________ of the feasible

region.

_________________________ is a mathematical technique used to determine which solutions in the feasible

region result in the optimal solutions of the objective function.

To determine the optimal solution to an optimization problem using linear programming we follow 5 steps:

1: Identify the quantity that must be optimized.

2: Define the variables that affect the quantity to be optimized and state any restrictions.

3: Write a system of linear inequalities to describe all the constraints of the problem and graph the

feasible solution. Graph the feasible solution.

4: Write the objective function.

5: Write the coordinates of the vertices of the feasible region. Test the coordinates of the vertices in

the objective function.

Ex. #1 A craft shop makes copper bracelets and necklaces. Each bracelet requires 15 minutes of cutting time

and 10 minutes of polishing time. Each necklace requires 15 minutes of cutting time and 20 minutes of

polishing time. There are a maximum of 225 minutes of cutting time and 200 minute of polishing time

available each day. The shop makes a profit of $5 on each bracelet and $7 on each necklace sold. How

many of each should the make per day to earn the most money?

Step 1: Identify the quantity that must be optimized.

Step 2: Define the variables that affect the quantity to be optimized and state any restrictions.

Step 3: Write a system of linear inequalities to describe all the constraints of the problem and graph the

feasible solution. Graph the feasible solution.

Step 4: Write the objective function.

Step 5: Write the coordinates of the vertices of the

feasible region. Test in Objective function.

Homework: pg. 341 #1, 4, 5, 11-15

Ex. #2 There are two different brands of lawn fertilizer:

Brand A

(kg per bag)

Brand B

(kg per bag)

Nitrogen 30 20

Phosphoric acid 2 4

Potash 1 4

A lawn needs at least 120 kg of nitrogen, 16 kg of phosphoric acid, and at least 12 kg of potash. Brand

A costs $22 per bag and Brand B costs $18 per bag. How many bags of each brand should be used to

minimize cost? What is the minimum cost?

Step 1: Identify the quantity that must be optimized.

Step 2: Define the variables that affect the quantity to be optimized and state any restrictions.

Step 3: Write a system of linear inequalities to describe all the constraints of the problem and graph the

feasible solution. Graph the feasible solution.

Step 4: Write the objective function.

Step 5: Write the coordinates of the vertices of the

feasible region. Test in Objective function.