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LINEAR PROGRAMMING |
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In solving linear programming problems, there are two stages:
1.
To
interpret the information given as a series of simultaneous inequalities and
display them graphically.
2.
To
investigate some characteristic of the points in the unshaded solution set.
Questions
For question 1 to 3 draw an accurate graph to
represent the inequalities listed, using shading to show the unwanted regions.
1. x + y ≤
11: y ≥ 3; y ≤
x
Find the point having whole number coordinates and
satisfying these inequalities which gives;
A)
the
maximum value of x + 4y
B)
the
minimum value of 3x + y
2. 3x +
2y ‹ 24;
x + y ‹ 12:
y › ½x; y ‹ 1.
Find the point having whole number coordinates and satisfying these
inequalities which gives;
a)
the
maximum value of 2x + 3y
b)
the
minimum value of x + y
3. 3x +
2y ≤ 60;
x + 2y ≤ 30;
x ≥ 10; y
≥ 0
a)
the
maximum value of 2x + y
b)
the
maximum value of xy
4.
A boy is given $1.20 to buy some peaches and
apples. Peaches cost 20c each, apples
10c each. He is told to buy at least 6
individual fruits, but he must not buy more apples than peaches.
Let x be the number of peaches
he buys.
Let y be the number of apples he
buys.
a)
Write
down three inequalities which must be satisfied.
b)
Draw
a linear programming graph and use it to list the combinations of fruit that
are open to him
5.
A
girl is told to buy some melons and oranges.
Melons are 50c each and oranges 25c each, and she has $2 to spend. She must not buy more than 2 melons and she
must buy at least 4 oranges. She is
also told to buy at least 6 fruits all together.
Let x be the number of melons
Let y be the number of oranges.
a)
Write
down four inequalities which must be satisfied.
b)
Draw
a graph and use it to list the combinations of fruit that are open to her.
6.
A
chef is going to make some fruit cakes and sponge cakes. He has plenty of all
ingredients except for flour and sugar.
He has only 2000g of flour and 1200g of sugar. A fruit cake uses 500g of flour and 100g of sugar. A sponge cake uses 200g of flour and 200g of
sugar. He wishes to make more than 4
cakes all together.
7.
A
man has a spare time job spraying cars and vans. Vans take 2 hours each and cars take 1 hour each. He has 14 hours available per week. He has an agreement with one firm to do 2 of
their vans every week. Apart from that
he has no fixed work.
His permission to use his back
garden contains the clause that he must be at least twice as many cars as vans.
Let x be the number of vans
sprayed each week
Let
y be the number of cars sprayed each week.
a)
Write
down three inequalities which must be satisfied.
b)
Draw
a graph and use it to list the possible combinations of vehicles which he can
spray each week.
2
8.
The
manager of a football team has $100 to spend on buying new players. He can buy defenders at $6 each or forwards
at $8 each. There must be at least 6 of
each sort. To cover for injuries he
must buy at least 13 players all together.
Let x represent the number of
defenders he buys and
Let y represent the number of
forwards.
a)
In
what ways can he buy players?
b)
If
the wages are $10 per week for each defender and $ 20 per week for each
forward, what is the combination of players, which has the lowest wage bill?
9.
A
tennis-playing golfer has $ 15 to spend on golf balls (x) costing $1 each and
tennis balls (y) costing 60c each. He
must buy at least 16 altogether and he must buy more golf ball than tennis
balls.
a)
What
is the greatest number of balls he can buy?
b)
After
using them, he can sell golf balls for 10c each and tennis balls for 20c
each. What is his maximum possible
income from sales?
10.
A
travel agent has to fly 1000 people and 35,000kg of baggage from London to
Paris. Two types of aircraft are
available: A which takes 100 people and
2000kg of baggage, or B which takes 60 people and 3000kg of baggage. He can use no more than 16 aircraft
altogether. Write down three
inequalities which must be satisfied if he uses x of A and y of B.
a)
What
is the smallest number of aircraft he could used?
b)
If
the hire charge for each aircraft A is $10,000 and each aircraft B is $12,000,
find the cheapest option available to him.
c)
If
the hire charges are altered so that each A costs $10,000 and each B costs
$20,000, find the cheapest option now available to him.
11.
A
farmer has to transport 20 people and 32 sheep to a market. He can use either Fiats (x) which take 2
people and 1 sheep, or Rolls Royces (y)
which take 2 people and 4 sheep. He
must not use more than 15 cars altogether.
a)
What
is the lowest total numbers of cars he could use?
b)
If
it costs $10 to hire each Fiat and $30 for each Rolls Royce. What is the cheapest solution?
12.
A
shop owner wishes to buy up to 20 televisions for stock. He can buy either type C for $150 each or
type B for $300 each. He has a total of
$4500 he can spend. He must have at
least 6 of each type in stock. If he
buys x of type A and y of type B, write down 4 inequalities which must be
satisfied and represent the information on a graph.
a)
If
he makes a profit of $40 on each of type A and $ 100 on each of type B, how
many of each should he buy for maximum profit?
b)
If
the profit is $80 on each of type A and $100 on each of type B. How many of each should he buy now?
13.
A
farmer needs to buy up to 5 cows for a new herd. He can buy either brown cows (x) at $50 each or black cows (y) at
$80 each and he can spend a total of no more than $1600. He must have at least 9 of each type.
On selling the cows he makes a
profit of $50 on each brown cow and $60 on each black cow. How many of each sort should he buy for
maximum profit?
3
14.
The
manager of a car park allows 10m2 of parking space for
each car and 30m2 for each lorry.
The total space available is 300m2. He decides that the maximum number of vehicles at any time must
not exceed 20 and he also insists that there must be at least as many cars as
lorries. If the number of cars is x and
the number of lorries is y, write down three inequalities which must be
satisfied.
a)
If
the parking charge is $1 for a car and $ 5 for a lorry, find how many vehicles
of each kind he should admit to maximize his income.
3
b)
If
the charges are changed to $2 for a car and $3 for a lorry, find how many of
each kind he would be advised to admit.
15.
A
factory manager is planning to buy two types of machines. Type X needs 3m2 of floor space,
type Y needs 2m2 and she has 40m2 available. The cost of type X is £20, of type Y £100
and she can spend up to £900. Find the
greatest number of machines she can buy.
16.
A
farmer is going to sow oats and barley.
He estimates that oats need 4 men per hectare, barley 6 men and he has
26 men available. Oats cost £12 per
hectare, barley £8 per hectare. And he is prepared to spend up to £48. Find the greatest possible area of land he can
sow.
17.
A
dealer is going to buy radio and television sets. She intends to buy a total of 100 sets. A radio set costs £40, a television set £120 and she is prepared to spend £10400. The profit on a radio set is £16, on a television
£32. Find the maximum profit she can
make.
18.
Two
foods A and B contain respectively 4 and 6 units of protein per kg and 5 and 3
units of starch per kg. The cost of A
is 40p a kg, of B is 50p a kg.
If the minimum daily intake is to be at least 16 units of protein and 11
units of starch, what is the cheapest way of meeting those conditions?
19.
A
shepherd is planning a sheepfold using an existing wall as one side of the
sheepfold. He will use hurdles 2m in
length and there will be no overlap of hurdles. The breadth of the sheepfold must be greater than 20m, and the
length greater than 40m. He has 45
hurdles. List the possible lengths and
breadths of the sheepfold, and find which has the greatest area.
20.
A
tobacconist proposes to buy up to 500 pipes.
He has the choice of two kinds, one at £1 each and the other at £3 each,
and he can spend up to £1100. The
profit on a £3 pipe is twice that on a £1 pipe. How many of each kind should he buy to make as large a profit as
possible?
21.
The
number of units of vitamins A and B per kg of two breakfast cereals X and Y are
shown in the table.
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Vitamin
A |
Vitamin
B |
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X |
16 |
8 |
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Y |
18 |
4 |
The minimum daily intake required is 120 units of A
and 40 units of B. What is the least
total weight of breakfast cereal a man must eat to have enough of these
vitamins?
22.
A
woman has 7 guests in her house and is prepared to spend £2 on papers for
them. The Financial Gazette costs 20p and The sporting Record costs 25p.
She wishes each guest to have at least one of these papers, but
obviously, no guest wants more than one copy of either paper. Six guests she knows insist on The Financial
Gazette. List all possible solutions to
her problem of how many of each paper to buy.
23.
A
factory manager wishes to install two types of machines, small and large. Small machines need 2 operators and occupy
4m2 of floor space: large machines need 3 operators and 8cm2
of floorspace. There are up to 56
operators available and 136m2 of floorspace. The profit per week is £3 on a small machine
and £5 on a large. Find the greatest
weekly profit.
24.
The
Post Office is planning to sell books of stamps containing only 6p and 8p
stamps. The cost of the book must not
exceed £2, and must contain at least 30 stamps. What is the greatest umber of 8p stamps there could be in the book? If the stamps are printed in multiples of
four, and there must be some 8p stamps in the book, how many of each can be
included in the book?