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 |

The objective function is, Z = 0.17

[Bond A yields 17% returns and bond B yields 22% returns, we have to maximize 0.17

[Sum of the investments.]

[Constraint on investment in bond A.]

[Constraint on investment in bond B.]

[Relation between the investments.]

[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 |

The objective function D = 15

[Profit on sweet A is $15 per pound and profit on sweet B is $20 per pound.]

[The pack should not exceed 7 pounds.]

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

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

[

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 |

[The plant capacity is 1600 bags per day.]

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

(0.48)

[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 = 8

[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 |

[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 |

[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 |

[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 |

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

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

Therefore the system of inequalities that models the given situation is

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 = 22$z$ + 14$a$ | ||

b. | C = 14$z$ + 22$a$ | ||

c. | C = 308$z$$a$ | ||

d. | C = 36 + $z$ + $a$ |

The cost C of

Correct answer : (2)

9.

What is the maximum value of the objective function C = 6$x$ + 5$y$ subject to the constraints $x$ ≥ 0, $y$ ≥ 0, $x$ + $y$ ≤ 4 ?

a. | 44 | ||

b. | 22 | ||

c. | 29 | ||

d. | 24 |

Constraints are

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 = 6

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 |

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

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

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

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

So, the system of inequalities that models the given situation is:

Correct answer : (4)