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Linear Programming Worksheet

Linear Programming Worksheet
  • Page 1
 1.  
A person wants to invest $20,000 in two types of bonds A and B. Bond A yields 17% returns and bond B yields 22% returns on the amount invested. After some consideration, he decides to invest at least $5,000 in bond A and not more than $8,000 in bond B. He also wants to invest at least as much in bond A as in bond B. Which of the following investments gives him maximum returns?
a.
$12,000 in bond A and $8,000 in bond B
b.
$10,000 in bond A and $10,000 in bond B
c.
$3,400 in bond A and $4,400 in bond B
d.
$5,000 in bond A and $8,000 in bond B


Solution:

Let $x be the amount invested in bond A and $y be the amount invested in bond B.

The objective function is, Z = 0.17x + 0.22y.
[Bond A yields 17% returns and bond B yields 22% returns, we have to maximize 0.17x + 0.22y.]

x + y ≤ 20,000
[Sum of the investments.]

x ≥ 5,000
[Constraint on investment in bond A.]

y ≤ 8,000
[Constraint on investment in bond B.]

xy or x - y ≥ 0
[Relation between the investments.]

x ≥ 0, y ≥ 0
[Investment can not be negative.]


The feasible region determined by the above constraints is shown.

From the figure the vertices are (5000, 0), (20000, 0), (12000, 8000), (8000, 8000) and (5000, 5000)
[Read the points from the corresponding red marks.]

At (5000, 0), Z = (0.17)(5000) + (0.22)(0) = 850.
[Substitute the values.]

At (20000, 0), Z = (0.17)(20000) + (0.22)(0) = 3400.
[Substitute the values.]

At (12000, 8000), Z = (0.17)(12000) + (0.22)(8000) = 3800.
[Substitute the values.]

At (8000, 8000), Z = (0.17)(8000) + (0.22)(8000) = 3120.
[Substitute the values.]

At (5000, 5000), Z = (0.17)(5000) + (0.22)(5000) = 1950.
[Substitute the values.]

So, investing $12000 in bond A, and $8000 in bond B gives maximum returns.


Correct answer : (1)
 2.  
A sweet shop makes gift packet of sweets which combines two special types of sweets A and B. The weight of the pack is 7 pounds. At least 3 pounds of A and no more than 5 pounds of B should be used. The profit on sweet A is $15 per pound and on sweet B is $20 per pound. Which of the following combination in the pack gives maximum profit for the shop?
a.
3 pounds of sweet A & 4 pounds of sweet B
b.
4 pounds of sweet A & 4 pounds of sweet B
c.
3 pounds of sweet A & 2 pounds of sweet B
d.
3 pounds of sweet A & 3 pounds of sweet B


Solution:

Let a pack of x pounds of sweet A, y pounds of sweet B gives maximum profit.

The objective function D = 15x + 20y
[Profit on sweet A is $15 per pound and profit on sweet B is $20 per pound.]

x + y ≤ 7
[The pack should not exceed 7 pounds.]

x ≥ 3.
[The pack should have at least 3 pounds of sweet A.]

y ≤ 5
[The pack should not have more than 5 pounds of sweet B.]

x ≥ 0, y ≥ 0
[x , y are the number of pounds.]


The feasible region determined by the above constraints is shown.

From the figure the vertices are (3, 0), (7, 0) and (3, 4).

At (3, 0), P = 15(3) + 20(0) = 45
[Substitute the values.]

At (7, 0), P = 15(7) + 20(0) = 105
[Substitute the values.]

At (3, 4), P = 15(3) + 20(4) = 125
[Substitute the values.]

P is maximum at (3, 4). So, a pack of 3 pounds of sweet A and 4 pounds of sweet B gives maximum profit.


Correct answer : (1)
 3.  
Ed owns a cement manufacturing plant where two types of cement, namely powdered and granulated are produced. His plant works 8 hours a day and has a maximum capacity to produce 1600 bags per day. He has to produce at least 600 bags of powdered cement per day to meet the demand. The time required to make a bag of powdered cement is 0.24 minutes which is half of the time required to produce granulated cement. He earns a profit of $8 per bag for granulated cement and $7 per bag for powdered cement. Find the maximum profit of Ed per day.
a.
$11800
b.
$9800
c.
$4200
d.
$11600


Solution:

Let x be the number of bags of granulated cement produced per day and y be the number of bags of powdered cement produced per day.

x + y ≤ 1600
[The plant capacity is 1600 bags per day.]

y ≥ 600
[The plant must produce at least 600 bags of powdered cement per day.]

(0.48) x + (0.24) y ≤ 8(60) or 2x + y ≤ 2000
[One bag of powdered cement takes 0.24 minutes, one bag of granulated cement takes 0.48 minutes and the plant works for 8 hours a day.]

The objective function is, the profit, P = 8x + 7y
[The profit on a bag of granulated cement is $8 and the profit on a bag of powdered cement is $7.]


The feasible region determined by the above constraints is shown.

From the figure the vertices are (0, 600), (700, 600), (400, 1200) and (0, 1600).

At (0, 600), P = 8(0) + 7(600) = 4200
[Substitute the values.]

At (700, 600), P = 8(700) + 7(600) = 9800
[Substitute the values.]

At (400, 1200), P = 8(400) + 7(1200) = 11600
[Substitute the values.]

At (0, 1600), P = 8(0) + 7(1600) = 11200
[Substitute the values.]

The maximum value of P is $11600. The maximum profit of Ed per day is $11600.


Correct answer : (4)
 4.  
Limits on the variables in the objective function are called________.
a.
extremes
b.
restrictions
c.
vertices
d.
objectives


Solution:

Limits on the variables in the objective function are called restrictions.
[Definition of the restrictions.]


Correct answer : (2)
 5.  
The graphical tool which identifies conditions that make as large as possible or as small as possible is known as_________.
a.
optimization technique
b.
curve sketching
c.
graph theory
d.
linear programming


Solution:

The graphical tool which identifies conditions that make as large as possible or as small as possible is known as linear programming.
[Definition of linear programming.]


Correct answer : (4)
 6.  
The quantity to be maximized or minimized is represented by a linear function called as _________.
a.
the constraint
b.
the feasible function
c.
the objective function
d.
the restriction


Solution:

The quantity to be maximized or minimized is represented by a linear function called the objective function.
[Definition of the objective function.]


Correct answer : (3)
 7.  
Josh wants to buy at least 13 books and at least 19 cakes from a shop. If y represents the number of books purchased and a represents the number of cakes purchased, then which of the following systems of inequalities models this situation?
a.
y ≥ 13, a ≥ 19
b.
y ≤ 13, a ≥ 19
c.
y ≥ 19, y ≤ 13
d.
y ≥ 13, a ≤ 19


Solution:

y represents the number of books purchased and a represents the number of cakes purchased.

The minimum number of books to be purchased is 13. So, y ≥ 13.

The minimum number of cakes to be purchased is 19. So, a ≥ 19.

Therefore the system of inequalities that models the given situation is
y ≥ 13, a ≥ 19.


Correct answer : (1)
 8.  
If the cost of a tape is $14 and the cost of a CD is $22, then write the objective function for the cost(C) of z tapes and a CDs .
a.
C = 22z + 14a
b.
C = 14z + 22a
c.
C = 308za
d.
C = 36 + z + a


Solution:

The cost of each tape is $14 and the cost of each CD is $22.

The cost C of z tapes and a CDs is C = 14z + 22a, which is the objective function for the cost.


Correct answer : (2)
 9.  
What is the maximum value of the objective function C = 6x + 5y subject to the constraints x ≥ 0, y ≥ 0, x + y ≤ 4?
a.
44
b.
22
c.
29
d.
24


Solution:

Objective function is C = 6x + 5y

Constraints are x ≥ 0, y ≥ 0, x + y ≤ 4.


The feasible region determined by the given constraints is shown in the graph.
[Draw the constraints.]

From the graph, the three vertices are (0, 0), (4, 0) and (0, 4).

To evaluate the minimum or maximum values of C, we evaluate C = 6x + 5y at each of the three vertices.

At (0, 0) , C = 6(0) + 5(0) = 0
[Substitute the values.]

At (4, 0) , C = 6(4) + 5(0) = 24
[Substitute the values.]

At (0, 4) , C = 6(0) + 5(4) = 20
[Substitute the values.]

So, the maximum value of the objective function, C is 24.


Correct answer : (4)
 10.  
Sunny can afford to buy 14 bananas and 19 apples. He wants at least 12 bananas and at least 16 apples. If x represents the number of bananas, z represents the number of apples that Sunny buys, then which of the following systems of inequalities models this situation?
a.
x ≥ 14, z ≥ 19, x ≥ 12, z ≥ 16
b.
x ≤ 12, z ≤ 16, x ≥ 14, z ≥ 19
c.
x ≤ 14, z ≤ 19, x ≤ 12, z ≤ 16
d.
x ≤ 14, z ≤ 19, x ≥ 12, z ≥ 16


Solution:

x represents the number of bananas and z represents the number of apples that Sunny buys.

The maximum number of bananas that can be bought is 14. So, x ≤ 14.

The maximum number of apples that can be bought is 19. So, z ≤ 19.

The minimum number of bananas to be bought is 12. So, x ≥ 12.

The minimum number of apples to be bought is 16. So, z ≥ 16.

So, the system of inequalities that models the given situation is:
x ≤ 14, z ≤ 19, x ≥ 12, z ≥ 16.


Correct answer : (4)

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