1. In each of the following, use arrows to indicate the graphical solution of each inequality on the same rectangular coordinate plane, then shade the region which represents the graphical solution of the system of inequalities.
(a) ���
≤+≥
32yx
x (b)
���
−−<+≥
12
xyxy
O
1
y
1 3 4x
3
2
−2
−1
x = 2
2
x + y = 3
O
1
y
−4 1 2x
3
2
−2
−1−3 −1
y = −x − 1
−2
y = x + 2
2. In each of the following, the shaded region / dots represent(s) the graphical solution of a system of inequalities. Write down the system of inequalities.
(a)
O
1
y
1 3 4x
3
2
−2
−1−2 −1
2x + 3y = 6
2
x = 3
y = 2
(b)
O
1
y
1 4 5x
−4
−3
−1
y = −x + 3
2
y = −2−2
−13
y = x − 3
The system of inequalities is The system of inequalities is
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�
.
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�
���
�
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era dna
yx
.
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The overlapping region of the graphical solutions of the inequalities is the graphical solution of the system of inequalities.
1. A shop sells two types of barbecue food sets, A and B. The number of chicken wings and number of sausages in each set A and each set B are as follows.
Chicken wing Sausage
Set A 6 4
Set B 5 9
It is given that Ken requires at least 50 chicken wings and at most 80 sausages. If x sets A and y sets B are to be bought, write down all the constraints about x and y.
The constraints are
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�
�
.
2. A carpenter is going to use at most 150 units of wood and 180 working hours to make x wardrobes and y beds. It is given that making a wardrobe requires 15 units of wood and 12 working hours, while making a bed requires 10 units of wood and 15 working hours.
1. In each of the following, the shaded region / dots represent(s) the feasible solutions of certain constraints. If ) ,( yx is any point in the feasible region, find the maximum and minimum values of
) ,( yxf .
(a) yxyxf += 2) ,(
O
21
y
1 2x
3 4−1−1
7
56
34
75 62x + y = 0
From the graph,
) ,( yxf attains its maximum / minimum values at the points ) , ( and ) , ( .
(b) Using the result of (a), find the maximum and minimum values of yxyxf 23) ,( −= subject to the above constraints.
Consider the vertices ) , ( , ) , ( and ) , ( of the feasible region.
At the point ) , ( ,
) , (
=
=f
At the point ) , ( ,
) , (
=
=f
At the point ) , ( ,
) , (
=
=f
∴ Maximum value =
Minimum value =
3. A company is going to spend at most $120 000 and 720 man-hours to produce x items A and y items B. The details of producing an item A and an item B are as follows.
(c) If the profits of selling an item A and an item B are $400 and $500 respectively, how many items A and items B should the company produce to obtain the maximum profit?
Total profit
$
) $() ,($
=
+= yxyxP
Draw a straight line , i.e. the straight line on the
graph in (b).
From the graph, the profit is the maximum when =x and =y .
In each of the following, the shaded region / dots represent(s) the graphical solution of a system of inequalities. Write down the system of inequalities. (5 − 6)
(b) It is given that (1, −1) and (3, 1) both satisfy the system of inequalities
���
���
�
<<≥−
<+≤−
kyhyx
yxyx
045
1032
, where h and
k are real numbers.
(i) Find the greatest integral value of h. (ii) Find the smallest integral value of k. (iii) At least how many integral solutions does the system of inequalities have?
27. Iris is organizing a birthday party and she is going to buy x L of cola and y L of orange juice according to the following constraints.
I. The total capacity of cola and orange juice should not be less than 10 L.
II. The total capacity of cola should be at least twice that of orange juice.
Write down all the constraints about x and y.
28. A factory is going to produce x toys P and y toys Q. The details of producing a toy P and a toy Q are as follows.
Material required (unit) Production time (hour)
Toy P 6 2
Toy Q 3 5
Given that there are 1 200 units of materials and 600 working hours available, write down all the constraints about x and y.
29. There are 480 apples, 240 mangoes and 300 oranges available in a fruit shop, and a shopkeeper is going to sell the fruits by packing them into fruit baskets A and fruit baskets B. The details of each type of fruit basket are as follows.
Apple Mango Orange
Fruit basket A 8 4 6
Fruit basket B 6 8 4
If x fruit baskets A and y fruit baskets B are to be produced, write down all the constraints about x and y.
30. A restaurant provides two types of soup, vegetable soup and seafood soup, each day. It is given that
the cost of each litre of vegetable soup and each litre of seafood soup are $5 and $10 respectively, and the restaurant prepares x L of vegetable soup and y L of seafood soup each day according to the following constraints.
I. The total cost of preparing the two types of soup each day should not be more than $600. II. At most 80 L of soup should be prepared each day.
(a) Write down all the constraints about x and y. (b) Represent the feasible solutions on a rectangular coordinate plane.
31. A grocery store sells toothpaste and toothbrushes in two types of packages, family set and economy set. The details of each type of package are as follows.
Number of tubes of toothpaste Number of toothbrushes
Family set 2 4
Economy set 4 6
Tommy wants to buy at least 20 tubes of toothpaste and 36 toothbrushes as gifts of an oral health promotional event. If x family sets and y economy sets are to be bought,
(a) write down all the constraints about x and y. (b) represent the feasible solutions on a rectangular coordinate plane.
32. A beverage shop has 900 pears and 600 apples available to prepare x jars of special drink A and y jars of special drink B. It is given that preparing a jar of special drink A requires 3 pears and 1 apple, while preparing a jar of special drink B requires 2 pears and 3 apples.
(a) Write down all the constraints about x and y. (b) Represent the feasible solutions on a rectangular coordinate plane.
33. An ice-cream shop provides two types of dessert sets, A and B. The details of each type of dessert sets are as follows.
Green tea ice-cream (mL)
Chocolate ice-cream (mL)
Mango ice-cream (mL)
Dessert set A 100 100 100
Dessert set B 100 200 0
It is given that at most 40 L of green tea ice-cream, 50 L of chocolate ice-cream and 28 L of mango ice-cream are available for preparing dessert set A and dessert set B each week. If x dessert sets A and y dessert sets B are prepared each week,
(a) write down all the constraints about x and y.
(b) represent the feasible solutions on a rectangular coordinate plane.
34. There are 1 600 units of chemicals A, 1 800 units of chemicals B and 1 500 units of chemicals C available in a laboratory for producing x mL of vaccine P and y mL of vaccine Q. The details of 1 mL of vaccine P and 1 mL of vaccine Q are as follows.
Chemical A (unit) Chemical B (unit) Chemical C (unit)
Vaccine P 4 6 4
Vaccine Q 5 4 5
(a) Write down all the constraints about x and y.
(b) Represent the feasible solutions on a rectangular coordinate plane.
35. An organization is going to invite x male guests and y female guests to a ball. The number of souvenirs and the amounts of food and drink prepared for each male guest and each female guest are as follows.
Souvenir Fruit tart Champagne (glass)
Male guest 1 3 2
Female guest 1 2 1
It is given that the organization has prepared 100 souvenirs, 240 fruit tarts and 150 glasses of champagne.
(a) Write down all the constraints about x and y.
(b) Represent the feasible solutions on a rectangular coordinate plane.
36. The owner of a pet shop is going to use at most $9 000 to buy some cats and dogs. Due to the limitation of space, the pet shop can only hold at most 45 cats and dogs in total, and the number of dogs should not be more than that of cats. It is given that the cost of each cat and each dog are $180 and $300 respectively.
(a) Write down all the constraints about the number of cats and the number of dogs to be bought. (b) Represent the feasible solutions on a rectangular coordinate plane.
37. A factory produces digital video discs (DVD) and video compact discs (VCD). For a box of DVD, $7 are spent on materials and $3 on packaging. For a box of VCD, $3 are spent on materials and $2 on packaging. It is given that the material cost and packaging cost of producing the two types of discs each hour should not be more than $2 100 and $1 200 respectively, and the number of boxes of DVDs produced each hour should not be more than 1.5 times that of VCDs.
(a) Write down all the constraints about the number of boxes of DVDs and the number of boxes of VCDs produced each hour.
(b) Represent the feasible solutions on a rectangular coordinate plane.
38. A train provides at most 240 m2 of floor area for passengers and carries at most 3 200 kg of luggage. There are two types of seats available, first-class seats and economy-class seats, where each of them occupies 3 m2 and 1.6 m2 of floor area respectively. It is given that the number of first-class seats
should be less than
41
of the number of economy-class seats, and at most 64 kg and 20 kg of luggage is
allowed for each passenger taking first-class seat and economy-class seat respectively.
(a) Write down all the constraints about the number of first-class seats and the number of economy-class seats provided.
(b) Represent the feasible solutions on a rectangular coordinate plane.
39. A factory has 300 kg of white rice, 175 kg of red rice and 250 kg of brown rice available to produce two rice mixtures A and B. The details of each kg of mixture A and mixture B are as follows.
White rice (g) Red rice (g) Brown rice (g)
Mixture A 625 250 125
Mixture B 250 250 500
It is given that the amount of mixture B produced should not be more than twice that of mixture A, and the amount of mixture B produced should be at least 200 kg.
(a) Write down all the constraints about the amount of mixture A and mixture B to be produced. (b) Represent the feasible solutions on a rectangular coordinate plane.
In each of the following, if (x, y) is any point in the shaded region, find the values of x and y such that ) ,( yxf attains its maximum and minimum values. (40 − 41)
40. yxyxf −=) ,( 41. yxyxf 2) ,( +=
O
21
y
1 2x
56
3 6−1 4 5−1
34
O
−3
y
−4x
In each of the following, the shaded region / dots represent(s) the feasible solutions of certain constraints. If (x, y) is any point in the feasible region, find the maximum and minimum values of ) ,( yxf . (42 − 45)
56. A factory is going to produce x items A and y items B. The details of producing an item A and an item B are as follows.
Item A Item B
Material cost ($) 80 120
Production time (hour) 20 20
Given that there are $7 200 available for the materials and 1 400 working hours,
(a) write down all the constraints about x and y.
(b) represent the feasible solutions on a rectangular coordinate plane.
(c) If the profits of selling an item A and an item B are $300 and $350 respectively, how many items of each type should the factory produce to obtain the maximum profit?
57. Jacky is organizing a party and he is going to buy x packs of chocolate in package A and y packs of chocolate in package B. The table below shows the details of a pack of chocolate in each package.
Number of pieces of chocolate with nuts
Number of pieces of milk chocolate
Package A 8 8
Package B 4 12
Given that at least 40 pieces of chocolate with nuts and 48 pieces of milk chocolate are required in the party,
(a) write down all the constraints about x and y.
(b) represent the feasible solutions on a rectangular coordinate plane.
(c) If the selling prices of a pack of chocolate in package A and package B are $36 and $21.6 respectively, how many packs of chocolate in each package should Jacky buy to minimize the expenditure?
58. A coordinator is going to take 240 performers to the performance venue by hiring x coaches A and y coaches B. It is given that each coach A can carry 16 passengers and each coach B can carry 48 passengers, and the coach hire company can arrange at most 8 drivers and 3 coaches A.
(a) Write down all the constraints about x and y.
(b) Represent the feasible solutions on a rectangular coordinate plane.
(c) If the rental for a coach A and a coach B are $200 and $800 respectively, how many coaches of each type should be hired so that the rental is kept at the minimum? Find the minimum rental.
59. A carpenter is going to produce x pieces of furniture A and y pieces of furniture B. The details of producing a piece of furniture A and a piece of furniture B are as follows.
Material required (unit) Production time (hour)
Furniture A 15 5
Furniture B 20 3
Given that there are 150 units of materials and 28 working hours available,
(a) write down all the constraints about x and y.
(b) represent the feasible solutions on a rectangular coordinate plane.
(c) If the profits of selling a piece of furniture A and a piece of furniture B are $150 and $90 respectively, how many pieces of furniture of each type should the carpenter produce to maximize the profit? Find the maximum profit.
60. A bakery sells pancakes and cakes. Making each kg of pancake requires 2 mangoes and 1 peach, while making each kg of cake requires 4 mangoes and 3 peaches. It is given that the bakery has at most 50 mangoes and 30 peaches available to make x kg of pancakes and y kg of cakes each day.
(a) Write down all the constraints about x and y.
(b) Represent the feasible solutions on a rectangular coordinate plane.
(c) If the profit of selling each kg of cake is 2.5 times the profit of selling each kg of pancake, how many pancakes and cakes (in kg) should be made each day to obtain the maximum profit?
61. The owner of a stationery shop is going to sell at most 120 ball-point pens and 30 correction pens by packing them into x packages A and y packages B. The table below shows the number of ball-point pens and correction pens in each type of package.
Ball-point pen Correction pen
Package A 5 1
Package B 6 2
(a) Write down all the constraints about x and y.
(b) Represent the feasible solutions on a rectangular coordinate plane.
(c) If the profits of selling a package A and a package B are in the ratio of 3 : 5, how many packages of each type should be produced to obtain the maximum profit?
62. A food supplier provides students lunch sets with two types of food, A and B. The table below shows the nutritional content of food A and food B.
Carbohydrates (unit) Protein (unit) Fat (unit)
Every 100 g of food A 30 10 4
Every 100 g of food B 24 20 6
It is given that each lunch set should contain at least 108 units of carbohydrates, at least 72 units of protein and at most 36 units of fat. Suppose each lunch set includes x g of food A and y g of food B,
(a) write down all the constraints about x and y.
(b) represent the feasible solutions on a rectangular coordinate plane.
(c) If the cost for every 100 g of food A and food B are $2.8 and $4 respectively, how much food A and food B (in g) should each lunch set include to minimize the cost? Find the minimum cost of each lunch set.
63. A factory is going to spend not more than $53 000 and 300 minutes of working hours in producing 9 000 electronic components. The details of producing every 100 electronic components by three production lines are as follows.
Cost required ($) Production time (minute)
Production line A 500 4
Production line B 600 3
Production line C 800 2
Let x, y and z be the number of electronic components to be produced by production lines A, B and C respectively.
(a) (i) Express z in terms of x and y.
(ii) Write down all the constraints about x and y.
(b) Represent the feasible solutions on a rectangular coordinate plane.
(c) To let production line C produce the greatest possible number of electronic components, how many electronic components should production lines A and B produce? Find the greatest possible number of electronic components to be produced by production line C.
64. Ivan spends x hours in doing part-time job A and y hours in doing part-time job B each day under the following constraints.
I. He should work at least 5 hours and at most 10 hours each day. II. The time spent on part-time job B should be at most twice that on part-time job A. III. The time spent on part-time job A should not be more than that on part-time job B.
(b) Represent the feasible solutions on a rectangular coordinate plane.
(c) It is given that the hourly wages of part-time job A and part-time job B are $75 and $55 respectively.
(i) How should Ivan allocate his working hours so that his daily income attains its maximum? Find his maximum daily income.
(ii) If Ivan works at most 8 hours each day instead, how should he allocate his working hours so that his daily income attains its maximum?
65. The owner of a bakery uses at most 30 kg of grade A flour, 80 kg of grade B flour and 140 kg of grade C flour to prepare two flour mixtures P and Q. Mixture P is prepared by mixing grade B flour and grade C flour in the ratio of 1 : 2. Mixture Q is prepared by mixing grade A, grade B and grade C flour in the ratio of 1 : 2 : 3. It is given that the owner prepares x kg of mixture P and y kg of mixture Q.
(a) Write down all the constraints about x and y.
(b) Represent the feasible solutions on a rectangular coordinate plane.
(c) Given that the profit of making bread with each kg of mixture Q is three times that with each kg of mixture P,
(i) how much mixture (in kg) of each type should the owner prepare to maximize the profit?
(ii) When the profit is at the maximum, how much flour (in kg) of each type is used to prepare the two flour mixtures?
8.29 29
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Worksheet 8A - I (page 8.1) 2. (a) 2 xy (b) 33 yx
Worksheet 8A - II (page 8.3)
2. (a)
23
632
yx
yx (b)
sregetni era dna 2
33
yxy
xyxy
Worksheet 8B (page 8.5)
1.
sregetni evitagen-non era dna 80945056
yxyxyx
2. (a)
sregetni evitagen-non era dna 60543023
yxyxyx
Worksheet 8C (page 8.7) 1. (a) Maximum value 15, minimum value 1
