Finite Mathematics MAT 141: Chapter 3 Notes Graphing Linear - - PowerPoint PPT Presentation

MAT 141 ‐ Chapter 3 1

Finite Mathematics MAT 141: Chapter 3 Notes

Linear Programming David J. Gisch

Graphing Linear Inequalities

Linear Inequalities Graphing with Intercepts

• Find the -intercept.

▫ Substitute 0 and solve for .

• Find the y-intercept.

▫ Substitute x 0 and solve for .

Example: Graph the equation using the intercepts. 2 4 12

MAT 141 ‐ Chapter 3 2

Graphing with Intercepts

Example: Graph the equation using the intercepts. 150 300 15,000

Graphing with Intercepts

Example: Graph the equation using the intercepts. 2 3 12

Inequalities

Graph the equation using the intercepts. 2 3 12

Inequalities

Graph the equation using the intercepts. 2 3 12

MAT 141 ‐ Chapter 3 3

Which way do you S hade

2 Methods.

• 1. Test a point.

▫ Substitute a point (not on the line) into the

• equation. If it results in a true inequality

shade that side, if not, shade the other side.

• 2. Solve for Y and follow the sign.

▫ , means you shade up ▫ , means you shade down

S

• lid Line or Dotted Line?
• If there is an equals sign then it is a solid line.
• If there is not an equals sign then it is a dotted line.

3 12 12

Graphing with Intercepts

Example: Graph the inequality. 6 15 30

S ystems of Linear Inequalities

• You graph all of the inequalities and where all of the

shaded regions overlap is the solution.

MAT 141 ‐ Chapter 3 4

S ystems of Linear Inequalities

Graph 1 2 3 12

Linear Inequalities

Example: Graph the system of inequalities.

• 2

1 2 8

Linear Inequalities

Example: Graph the system of inequalities.

• 800 2000 400

Linear Inequalities

Example: Graph the system of inequalities.

• 2 5 10

2 8

MAT 141 ‐ Chapter 3 5

Feasible Region

Example: The Gillette company produces two popular electric razors, the M3Power™ and the Fusion Power ™. Due to demand, the number of M3Power razors is never more than half the number of Fusion Power razors. The factories production cannot exceed more than 800 razors per day. (a) Write a system of inequalities to express the conditions of the Gillette Company. (b) Graph the feasible region. Solving Linear Programming Problems Graphically

Feasible Region

• As briefly mentioned in the previous section, the graph
• f a system of linear inequalities is called referred to as a

feasible region.

▫ All the points in that region are scenarios that meet the limitations of our constraints.

MAT 141 ‐ Chapter 3 6

Linear Programming

• Linear Program m ing is a method for solving

problems in which a particular quantity that must be maximized or minimized is limited by other factors, called constraints.

• An objective function is an algebraic expression in

two or more variables describing a quantity that must be maximized or minimized.

Equations with Multiple Variables

• If an equation has two variables it is an equation that can

be graphed in the plane, hence 2D.

▫ If the variables have no powers then it is a linear equation. ▫ Example, 5 10 or 5 10

• If an equation has three variables it is an equation that can

be graphed in space, hence 3D.

▫ If the variables have no powers then it is a plane (flat surface). ▫ Example, z 3 12 or , 3 12

Obj ective Function

Example: Bottled water and medical supplies are to be shipped to victims of an earthquake by plane. Each container of bottled water will serve 10 people and each medical kit will aid 6 people. Let x represent the number

• f bottles of water to be shipped and y the number of

medical kits. Write the objective function that describes the number of people that can be helped.

Constraints

Example: Each plane can carry no more than 80,000

• pounds. The bottled water weighs 20 pounds per

container and each medical kit weighs 10 pounds. Let x represent the number of bottles of water to be shipped and y the number of medical kits. (a) Write an inequality that describes this constraint. (b) Are there any other constraints?

MAT 141 ‐ Chapter 3 7

Feasible Region

Example: Graph the feasible region from example 2.

Obj ective Functions and Feasible Regions

• The feasible region is the inputs, that fit our constraints,

for the objective function.

• The lowest point

(minimum) or highest point (maximum) of the graph of the

• bjective function

will be at a corner.

Feasible Region

Example: What are the corner points of the region in example 2?

20 10 80,000

Check the Points

Corner Points Objective Function , 0, 0 10 0 6 0 0 0, 8000 10 0 6 8000 48,000 4000, 0 10 4000 6 0 40,000

While it seems odd, we can help the most people if we ship zero containers of water and 8,000 medical kits. Of course if we wanted to include some minimum amount of water that would add another constraint and change our feasible region.

MAT 141 ‐ Chapter 3 8

S teps for Linear Programming

Applications of Linear Programming

Applications of Linear Programming

• In the last section we had few constraints and therefore

the corner points were easy to find.

• We now will look at having several constraints and using

systems of equations to find the corner points.

Did Y

• u Know?

MAT 141 ‐ Chapter 3 9

Flower Arranging

Example: Flowers Unlimited has two spring floral arrangements, the Easter Bouquet and the Spring

• Bouquet. The Easter Bouquet requires 10 jonquils and 20

daisies and produces a profit of $1.50. The Spring Bouquet requires 5 jonquils and 20 daisies and yields a profit of $1. How many of each type of arrangements should the florist make to maximize the profit if 120 jonquils and 300 daisies are available? (a) Write an objective function. , 1.5 1

Flower Arranging

(b) Write down the constraints.

• 10 5 120

20 20 300 (c) Graph the feasible region.

Flower Arranging

(d) Find the corner points.

Flower Arranging

(e) Substitute the corner points into the objective function.

MAT 141 ‐ Chapter 3 10

Flower Arranging

Example: A construction company needs to hire at least 100 employees for a project. They will need at least 30 more unskilled laborers than skilled laborers. At least 20 skilled laborers should be hired. The unskilled laborers earn $8 per hour, and the skilled laborer earns $15 per

• hour. How many employees should the company hire to

minimize its hourly cost while satisfying all of the requirements? (a) Write an objective functions. , 8 15

Flower Arranging

(b) Write down the constraints. 100 30 20 (c) Graph the feasible region.

Flower Arranging

(d) Find the corner points.

Flower Arranging

(e) Substitute the corner points into the objective function.

MAT 141 ‐ Chapter 3 11

Flower Arranging

Example: Certain animals at a rescue shelter must have at least 30 g of protein and at least 20 g of fat per feeding

• period. These nutrients come from food A, which cost 18

cents per unit and supplies 2 g of protein and 4 g of fat; and food B, which cost 12 cents per unit and supplies 6 g

• f protein and 2 g of fat. Food B is bought under a long

term contract requiring at least 2 units of B be used per

• serving. How much of each type of food must be bought to

minimize the cost per serving. (a) Write an objective functions. , .18 .12

Flower Arranging

(b) Write down the constraints.

• 2 6 30

4 2 20 2 (c) Graph the feasible region.

Flower Arranging

(d) Find the corner points.

Flower Arranging

(e) Substitute the corner points into the objective function.

MAT 141 ‐ Chapter 3 12

Flower Arranging

Example: At the end of every month, after filling orders for its regular customers, a coffee company has some pure Colombian coffee and some special-blend coffee remaining. The practice of the company has been to package a mixture of the two coffees into 1-pound packages as follows: a low-grade mixture containing 4 ounces of Colombian coffee and 12 ounces of special-blend coffee and a high-grade mixture containing 8 ounces of Colombian and 8 ounces of special-blend

• coffee. A profit of $0.30 per package is made on the low-grade

mixture, whereas a profit of $0.40 per package is made on the high- grade mixture. This month, 120 pounds of special-blend coffee and 100 pounds of pure Colombian coffee remain. How many packages of each mixture should be prepared to achieve a maximum profit? Assume that all packages prepared can be sold.

(a) Write an objective functions. , 0.30 0.40

Flower Arranging

(b) Write down the constraints. 4 8 1600 12 8 1920 (c) Graph the feasible region.

Flower Arranging

(d) Find the corner points. (e) Substitute the corner points into the objective function.

Insurance

Example: A company is considering two insurance premium plans with the types of coverage and premiums shown in the following figure. The company wants at least $300,000 fire/theft insurance and at least $3,000,000 liability insurance from these plans. How many of each type should they buy to minimize the cost?

(a) Write an objective functions. , 50 40

*This means $50 buys you one unit of Policy A, giving you $10,00 fire/theft coverage and $180,000 liability coverage.

MAT 141 ‐ Chapter 3 13

Flower Arranging

(b) Write down the constraints. 10,000 15,000 300,000 / 180,000 120,000 3,000,000 (c) Graph the feasible region.

Flower Arranging

(d) Find the corner points.

Flower Arranging

(e) Substitute the corner points into the objective function.

Insurance

Example: The Ric Shaw Chair company makes two types of rocking chairs, a plain chair and a fancy chair. Each rocking chair must be assembled and then finished. The plain chair takes 4 hours to assemble and 4 hours to finish. The fancy chair takes 8 hours to assemble and 12 hours to finish. The company can provide at most 160 worker-hours of assembly and 180 worker-hours of assembly per

• day. If the profit on a plain chair is $25 and the profit on a fancy chair

is $40, how many of each type should the company make to maximize profit?

(a) Write an objective functions. , 25 40

MAT 141 ‐ Chapter 3 14

Flower Arranging

(b) Write down the constraints. 4 8 160

• 4 12 180
• (c) Graph the feasible region.

Flower Arranging

(d) Find the corner points. (e) Substitute the corner points into the objective function.