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Skill Builder

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2CHAPTER

Skill Builder: From Words to Inequalities in Two Variables 67

1 Write the inequality that describes the following situation:

A box of cookies contains chocolate cookies and oatmeal cookies. The box contains a maximum of 30 cookies.

Let “x” represent the number of chocolate cookies and let “y” represent the number of oatmeal cookies.


A bouquet of flowers contains roses and carnations. The bouquet must contain a minimum of 12 flowers.

Let “x” represent the number of roses and let “y” represent the number of carnations.


At a local florist shop, yellow roses cost $2 and red roses cost $3. Steven wants to spend no more than $25 on a bouquet that includes both yellow and red roses.

Let “x” represent the number of yellow roses and let “y” represent the number of red roses.


A dance club hires two different DJ’s to play music on a Friday night. The hip hop DJ charges $100 per hour, while the rap DJ charges $50 per hour. The club can spend a maximum of $800, and they want both types of music played.

Let “x” represent the number of hours of hip hop and let “y” represent the number of hours of rap.


Quincy has, at most, $10 more than Kelly.

Let “x” represent the amount of money Quincy has and let “y” represent the amount of money Kelly has.


At a hobby store, the owner decides that he must display at least 20 more baseball cards than hockey cards.

Let “x” represent the number of hockey cards displayed and let “y” represent the number of baseball cards displayed.

1 From Words to Inequalities in Two Variables EVALUATED

Linear Programming

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68 Chapter 2: Linear Programming


There are at most twice as many blue marbles as red marbles.

Let “x” represent the number of blue marbles and let “y” represent the number of red marbles.


John spends at least 3 times as many hours playing piano as he does playing guitar.

Let “x” represent the number of hours John plays piano and let “y” represent the number of hours he plays guitar.

9 Duane is going to buy flowers for his wife. The local florist sells roses for $2 and carnations for $1. Duane’s bouquet must satisfy the following constraints:

• It contains at least 12 flowers. • It costs a maximum of $35. • There are at least 4 more roses than

carnations.

Let “x” represent the number of roses and let “y” represent the number of carnations. Determine the system of constraints that describes the situation.

10 A high school has organized a talent show. Two types of tickets have been sold: tickets for students and tickets for guests. The constraints associated with the sale of these tickets are represented as follows:

x + y ≤ 400 x ≥ 3y

where “x” is the number of student tickets sold and “y” is the number of guest tickets sold.

Write the above constraints in words.

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Skill Builder: Graphing a Polygon of Constraints 69

1 Graph the following system of inequalities on the Cartesian plane to make a polygon of constraints and identify its vertices.

0056

xyxy

≥≥≤≤

5 10x

5

10

0

y


0064

xyxy

≥≥<<

5 10x

5

10

0

y


0026

xyy xy

≥≥≥≤

5 10x

5

10

0

y


00

39

xy

y xx

≥≥≤

≤

5 10x

5

10

0

y

2 Graphing a Polygon of Constraints EVALUATED

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00

2 3 18

xy

x y

≥≥+ ≤

5 10x

5

10

0

y


00

83 5 15

xyx y

x y

≥≥+ ≤+ ≥

5 10x

5

10

0

y

7 Draw the polygon of constraints on the Cartesian plane that is associated with the following system of inequalities and identify its vertices.

002

4 3 24

xyy x

x y

≥≥≥+ ≤

5 10x

5

10

0

y

8 Draw the polygon of constraints on the Cartesian plane that is associated with the following system of inequalities and identify its vertices.

00

103 9

1

xyx y

x yy x

≥≥+ ≤+ ≥≤ +

5 10x

5

10

0

y

Skill Builder


Skill Builder: Graphing a Polygon of Constraints 71

9 Graph the following system of inequalities on the Cartesian plane to make a polygon of constraints. Identify which vertices are solutions.

00

5010

xyx yx y

≥≥+ <+ ≥

25 50x

25

50

0

y

10 Graph the following system of inequalities on the Cartesian plane to make a polygon of constraints. Identify the vertex that is a solution.

00

2 806

xyx y

y x

≥≥+ <≥

50 100x

50

100

0

y

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1 The system of inequalities and the polygon of constraints below are associated with the same optimization situation. Each side of the polygon and its corresponding inequality are identified by the same number. What are the coordinates of the vertex A for this polygon of constraints?

2≥x7+ ≤x y

1≥y

1

2

31 2

3A

2 The system of inequalities and the polygon of constraints below are associated with the same optimization situation. Each side of the polygon and its corresponding inequality are identified by the same number. What are the coordinates of the vertex B for this polygon of constraints?

6+ ≥x y5≤y5≤x

1

2

31

2

3

B


3+ ≥x y5 2 15≤ +y x2 5 15≥ −y x

1

2

3

1

2

3A

4 The system of inequalities and the polygon of constraints below are associated with the same optimization situation. Each side of the polygon and its corresponding inequality are identified by the same number. What are the coordinates of the vertex C for this polygon of constraints?

0≥x6+ ≤x y

0≥y

1

2

33+ ≥x y4

12

3

4

C

3 Finding the Vertices for a Polygon of Constraints EVALUATED

Skill Builder


Skill Builder: Finding the Vertices for a Polygon of Constraints 73

5 The system of inequalities and the polygon of constraints below are associated with the same optimization situation. Each side of the polygon and its corresponding inequality are identified by the same number. What are the coordinates of the vertex B for this polygon of constraints?

2≥x1 52

≤ − +y x

1

2

4 2≥ +y x31

2

3

B


1 2

3

1≤ +y x

9+ ≤x y

1

2

2≥y3

C


12

3

3≥x8+ ≤x y

1

2

3 ≥y x3

C


2≥x2 10+ ≤x y

1

2

4 2≥ +y x3

2

3

1C

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6+ ≥x y

5 18≤ +y x

1

2

2 9≥ −y x33

1

2

C


3

1

2

2 9+ ≥x y

2 5 29+ ≤x y

1

2

2 3 5≤ +x y3

A

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Skill Builder: Finding The Optimal Value 75

1 A polygon of constraints representing the solutions for a system of inequalities is drawn in the Cartesian plane below. The coordinates of the vertices of this polygon are also shown on the graph. Which of the vertices represent a solution for the system of inequalities?

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2 The solutions for a system of inequalities are represented in the Cartesian plane below. Points A, B and C are shown on the graph. Which of the points A, B and C represent solutions for this system of inequalities?

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4 Finding The Optimal Value EVALUATED

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3 The vertices of a polygon of constraints are shown on the graph below. If the function to be optimized is z = 2x + 3y – 10, which ordered pair maximizes this function?

My Calculations

4 The polygon of constraints below takes into account the operating constraints of a restaurant. The weekly costs involved in running the restaurant are calculated using the expression 9x + 7y + 90. What is the minimum weekly cost of running this business?

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5 A book will have between 25 and 40 pages. At least 10 of these pages will have photographs. The printer charges $2 to print a page with photographs and $1 for a page without. If x represents the number of pages with photographs and y represents the number of pages without, what is the function to be optimized to minimize the costs?

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6 An outdoor centre rents two types of boats: kayaks and canoes. The centre rents at least 20 boats per day and a maximum of 13 canoes. A kayak is rented for $15 per day and a canoe for $10 per day. If x represents the number of kayaks rented per day and y represents the number of canoes rented per day, what is the function that can be used to maximize the revenue for the outdoor centre?

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7 The polygon of constraints below takes into account the operating constraints of a day camp. Each senior counsellor is paid $150 per day, while every junior counsellor is paid $100 per day. What is the minimum daily cost of running this day camp?

x: # of senior counselors y: # of junior counselors

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8 Jill makes dresses and coats. There are different constraints on the number of dresses and coats she can make. The polygon of constraints below is associated with this situation. Points A, B, C and D are the vertices of the polygon of constraints. The revenue per dress is $140, and the revenue per coat is $150. How many dresses and how many coats must Jill make in order to maximize her revenue?

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9 The following polygon of constraints represents the solutions for an optimization situation that involves minimizing the cost of an order of sugar and salt. The values in the table below were calculated in order to determine the minimum cost. In this situation, how many solutions minimize the cost?

x: pounds of salt y: pounds of sugar

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10 In order to raise money for their class trip, the grade 11 students at Ashbury College are selling hats and shirts.

The expected profit per student is calculated according to the function

P = 5x + 10y

where x represents the number of hats sold per student and y represents the number of shirts sold per student.

The constraints associated with this situation are represented on the graph below. What are all the possible ordered pairs that yield a maximum profit per student?

My Calculations

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Skill Builder: Solving Optimization Problems Level 1 85

1 The CN tower in Toronto offers elevator rides to the top of the tower. Children’s tickets cost $8 each and adult tickets cost $12 each. On any given ride there will be at least 10 children and 5 adults. The elevator has room for a maximum of 45 passengers and can hold up to 1 680kg. The average child weighs 28kg and the average adult weighs 70kg. What is the maximum revenue the CN tower can make on one ride given the information below.

x : # of children’s tickets sold y : # of adult’s tickets sold

≥ 0x≥ 0y

≥ 10x≥ 5y

+ ≤ 45x y

+ ≤28 70 1680x y

My Calculations

2 Joe Rockhead Jewelers is a producer of rings and necklaces. Because of its small size, it can only produce 80 items each week. To meet certain conditions in its workshop, it must make at least 45 rings and at least 10 necklaces weekly. To meet it’s customers demand, it must make at least 3 times as many rings as necklaces. For each necklace and ring that is made, J.R. Jewelers earns a profit of $200 and $150 respectively. Using the information given below, determine the maximum weekly profit.

System of Constraints for J.R. Jewelers weekly jewelry production:

x : # of necklaces y : # of rings

≥ 0x≥ 0y

≥ 10x≥ 45y

+ ≤ 80x y

≥ 3y x

My Calculations

5 Solving Optimization Problems Level 1 EVALUATED

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3 Lawton works at two different jobs. At the first job, he works a minimum of 10 hours a week and at the second, a maximum of 40 hours per week. He must work at least 30 hours per week but no more than 60 hours per week. He must work at least as many hours at the second job as he does at the first. He makes $20 an hour at the first job and $25 an hour at the second job. What is Lawton’s maximum possible salary if

x : # of hours per week at 1st job y : # of hours per week at 2nd job

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4 Tanya must decide how many people are needed to be hired to work the lunchtime and suppertime shifts at her restaurant. There must be at least 18 people working during the two shifts. Tanya knows however that she cannot afford more than 22 people working those shifts. The lunchtime rush requires a minimum of 6 people working. For the suppertime rush, the number of employees must be at least two more than the number who work the lunchtime shift. The people that work the lunchtime shift make $8 per shift while the people that work the suppertime shift make $10 per shift. How many people should Tanya hire for each shift to minimize her costs given the above constraints?

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5 A football team wants to minimize its transportation cost for an away game. The following polygon of constraints represents the solutions for this situation.

# of vans

# of cars

A (1, 9)

B (7, 7)

C (9, 3)

D (5, 1)

It costs $100 to rent each van and $75 to rent each car.

On the day of the game, it is revealed that no more than 4 cars are available to rent. Because of this, the minimum transportation cost will increase. By how much will this minimum transportation cost increase?

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6 Dan works two different jobs at a restaurant. Each week, he divides his time between cooking and cleaning. The polygon of constraints below shows the different constraints that Dan faces.

x : # of hours cooking y : # of hours cleaning

A (5, 35)

B (20, 5)C (15, 5)

D (5, 15)

Dan is told that from now on the number of hours he spends on cleaning must be less than or equal to the number of hours he cooks. By how many dollars does this constraint decrease Dan’s maximum possible weekly income if he makes $20 an hour for cooking and $16 for cleaning?

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7 Different constraints limit the number of adults and children that can take a balloon ride. The following polygon of constraints shows the possible combinations of adults and children.

A (3, 5)

C (10, 2)

B (7.5, 7.5)

Ms. Sanders, the balloon captain, is paid $40 for each adult and $30 for each child that rides in the hot air balloon.

How many adults and children must ride in the hot air balloon to maximize Ms. Sanders’ revenue?

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8 Rosie is asked to put x dimes and y quarters into a piggy bank. She must, however, follow the constraints given below:

≥ 1x ≥ 1y + ≤0.10 0.25 1x y

How many combinations of dimes and quarters satisfy this situation?

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9 Terry is in charge of buying the food for a local fundraiser. She needs to buy enough food to be able to make the following snacks:

• At most twice as many brownies as cookies.

• A minimum of 100 but not more than 400 brownies.

• At least 200 cookies.

• At most 600 brownies and cookies.

Her cost is $0.45 per cookie and $0.75 per brownie. She will sell the cookies for $1 each and the brownies for $1.50 each. Given the situation described above, how many cookies and brownies does Terry need to sell in order to make the greatest profit?

Let x : # of cookies y : # of brownies

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10 Benwell Gaming makes video games for children. The two most popular games are the Unicorns and Rainbows games and the Hug a Bunny Rabbit game. Mr. Benwell knows from past experience that they sell at most twice as many Hug a Bunny Rabbit games as Unicorns and Rainbows games. Benwell Gaming cannot have more than 1500 of these games in stock. It costs $5 to produce the Unicorns and Rainbows game and $10 for the Hug a Bunny Rabbit game. Benwell Gaming spends at least $10 000 to produce these games. What is the maximum profit Benwell Gaming can make from selling these games if the Unicorns and Rainbows game sells for $35 and the Hug a Bunny Rabbit game sells for $50?

Let x : # of Unicorns and Rainbows games in stock y : # of Hug a Bunny Rabbit games in stock

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1 The Ride This skate board factory produces Mega skateboards and Supreme skateboards. Their maximum output is 40 skateboards a day and they must produce at least 20 Mega and 5 Supreme boards per day. Also, they must produce at least 3 times as many Mega boards as Supreme skateboards. The profit on a Mega skateboard is $55 and on a Supreme is $75. What is the factory’s maximum daily profit?

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2 Every year, the cubs sell calendars to raise money. Two types of calendars are sold: wall calendars and desk calendars. They sell at least twice as many wall calendars as desk calendars with a maximum of 500 wall calendars. They sell at least 300 and up to a maximum 600 calendars. The profit on a wall calendar is $4 and on a desk calendar it is $3. How many calendars of each type must the cubs sell to maximize their profit?

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6 Solving Optimization Problems Level 2 ENRICHMENT

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3 John wants to buy new lights for an outdoor skating rink. He estimates that the job requires $2500 worth of material and a minimum of 30 hours of labour. He hires an electrician and his apprentice. The electrician charges $20 per hour and the apprentice, $12 per hour. The local municipality requires that an apprentice not work for more hours than the electrician, but the apprentice must put in a minimum of 8 hours work. How many hours should the electrician and the apprentice work in order to minimize costs?

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4 The Yum-Yum-Gum-Gum chewing gum company makes two types of gum: Happy Chew and Lemon Chew. To make their superb gum, Yum-Yum has a two stage production. Each stick of Happy Chew requires 4 seconds at stage one and 6 seconds at stage two. Each stick of Lemon Chew requires 4 seconds at stage one and 2 seconds at stage two. Both stages run for a maximum of 8 hours a day. Also, they must make at least 2520 sticks of each type of gum per day. The profit on each stick of Happy Chew is 7 cents and the profit on each stick of Lemon Chew is 9 cents. Find the maximum daily profit of the Yum-Yum-Gum-Gum chewing gum company.

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5 The organizing committee of a county fair wants a program printed. A printer asks 10 cents for each page printed in black and white and 30 cents for each page in colour. The printer’s cost of producing the binding for one program is $1. The program must have no less than 30 pages, no more than 90 pages and at least 5 pages in colour. Because of pricing standards, the number of pages in black and white increased by 3 times the number of pages in colour cannot exceed 180 pages. The printer wants to maximize his profits. Find some suggestions that he can make to the organizing committee that would enable him to achieve his goal.

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6 Sharon sells carnations for Mother’s day. She has two prices: a small carnation for 50 cents and a large carnation for $1. She knows that she will not be able to sell more than 100 carnations, including no more than 60 large carnations. She predicts that the number of sales of the small ones will not be more than double the number of the large ones. She wishes to sell at least $40 worth of flowers. She receives a commission of 10 cents on a small carnation and 25 cents on the large one. What is the most amount of money she can make from commissions?

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7 To make some money for the summer, Mary decides to build birdhouses and flower planters. Each birdhouse takes 0.8m2 of wood to build and each planter takes 1m2 of wood. It takes 2 hours to build to a birdhouse but only 50 minutes for a planter. Mary has at most 100m2 of wood available and can devote a maximum of 150 hours to this project. If she can sell a birdhouse for $10 and a planter for $8, what is the most amount of money she can make?

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8 Sheila knows that the minimum daily requirements for her dog’s diet are 18 units of protein and 30 units of carbohydrates. Pet dog food costs $1.65 a can, contains 2 units of protein and 6 units of carbohydrates. Budget dog food costs $1.35 a can, contains 2 units of protein and 2 units of carbohydrates. What combination of the two brands of dog food should Sheila give her dog each day to ensure it has a proper diet at minimum cost?

My Calculations

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9 Sam is an accountant for EN Mines Ltd. The company has recently received the following order: at least 100 tonnes of high grade mineral, 160 tonnes of medium grade, and 200 tonnes of low grade. The daily production of the first mine is 1 tonne of high grade mineral, 2 tonnes of medium grade and 4 tonnes of low grade. The second mine produces 2 tonnes of each type daily. The first mine has expenses of $2500 per day for its mining expenses and the second costs $2000 per day. How many days should each mine be in operation to fill this order at the lowest cost?

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10 Samantha is an accountant for the EN Tent Company. The manufacturing process requires three main machines: A, B, and C. To manufacture a deluxe tent, machine A must be used for 4 hours, machine B for 2 hours, and machine C for 1 hour. To manufacture a standard tent, machine A must be used for 1 hour, machine B for 2 hours, and machine C for 2 hours. Machine A is available for a maximum of 20 hours a day, machine B 16 hours, and machine C 14 hours. The profit on a deluxe tent is $120 and $145 on a standard tent. What is the maximum daily profit?

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chapter linear programming 2 - mianh.pbworks.com

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