FOM11



FOM11

Chapter 6 Review

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. When you graph a linear inequality, the first thing you must do is draw the boundary line.

What is the equation of the boundary line for the linear inequality 2x + 2y < 16?

|a. |y = 8 – x |

|b. |x = 8 – y |

|c. |y = 16 – x |

|d. |y = 4 – 2x |

____ 2. What is the boundary line for the linear inequality 3x – 6y < 18?

|a. |y = [pic]x – 1 |

|b. |y = [pic]x – 6 |

|c. |y = [pic]x – 3 |

|d. |y = [pic]x – 2 |

____ 3. Describe the boundary lines for the following system of linear inequalities:

[pic]

|a. |Dashed line along y = x + 2; dashed line along y = –x |

|b. |Dashed line along y = x + 2; solid line along y = –x |

|c. |Solid line along y = x + 2; dashed line along y = –x |

|d. |Solid line along y = x + 2; solid line along y = –x |

____ 4. Which point is in the solution region for the following system of linear inequalities?

[pic]

|a. |(1, 2) |

|b. |(2, –1) |

|c. |(–10, 0) |

|d. |(–1, –1) |

____ 5. A football stadium has 60 000 seats.

• 70% of the seats are in the lower deck.

• 30% of the seats are in the upper deck.

• At least 40 000 tickets are sold per game.

• A lower deck ticket costs $100, and an upper deck ticket costs $60.

Let x represent the number of lower deck tickets.

Let y represent the number of upper deck tickets.

How would you write the objective function for revenue, R?

|a. |R = 70y + 30x |

|b. |R = 100x + 60y |

|c. |R = 70x + 30y |

|d. |R = 100y + 60x |

____ 6. Which location best describes where would you find the optimal solutions to an objective function?

|a. |outside the solution region |

|b. |at or near the points of intersection |

|c. |within the solution region |

|d. |along a boundary line |

____ 7. Which point in the model below would result in the maximum value of the objective function W = 5y – 10x?

[pic]

|a. |B (1, 4) |

|b. |C (4, 1) |

|c. |A (–2, 1) |

|d. |D (1, 2) |

____ 8. Where might you find the maximum solution to the objective function?

Restrictions:

x [pic] R

y [pic] R

Constraints:

–2 [pic] x [pic] 4

–2 [pic] y [pic] 4

Objective function:

N = 2x – y

|a. |(–2, –2) |

|b. |(4, 4) |

|c. |(4, –2) |

|d. |(–2, 4) |

____ 9. Audrey notices the number of people and dogs in a dog park.

• There are more people than dogs.

• There are at least 12 dogs.

• There are no more than 40 people and dogs, in total.

Let d represent the number of dogs and let p represent the number of people.

Which inequality represents one of the conditions on d and p based on the given information?

|a. |d – p [pic] 40 |

|b. |d – p [pic] 12 |

|c. |d < p |

|d. |2d [pic] p |

____ 10. A football stadium has 60 000 seats.

• 70% of the seats are in the lower deck.

• 30% of the seats are in the upper deck.

• At least 40 000 tickets are sold per game.

• A lower deck ticket costs $100, and an upper deck ticket costs $60.

Let x represent the number of lower deck tickets.

Let y represent the number of upper deck tickets.

What are the restrictions on x and y?

|a. |x ∈ W, y ∈ W |

|b. |x ∈ I, y ∈ I |

|c. |x ∈ R, y ∈ R |

|d. |No constraints. |

Short Answer

11. Graph the solution set for the linear inequality 3y – 6x < –1.

12. Determine two valid solutions for the following system of linear inequalities.

[pic]

13. Complete the graph of the solution set for the following system of inequalities.

{(x, y) | y > 1, x > –4}

[pic]

14. Graph the solution set for the following system of inequalities.

{(x, y) | y > x – 5, y [pic] 4, x [pic] R, y [pic] R}

15. The following model represents an optimization problem. Determine the maximum solution.

Restrictions:

x [pic] R

y [pic] R

Constraints:

x [pic] 0

y [pic] 0

2x + y [pic] 10

x + y [pic] 20

Objective function:

Q = 2y – 10x

16. April notices the number of people and dogs in a dog park.

• There are more than twice as many people as dogs.

• There are at least 10 dogs.

• There are no more than 50 people and dogs, in total.

What are the maximum and minimum number of legs at the dog park?

Problem

17. A student council is ordering signs for the autumn dance. Signs can be made in letter size or poster size.

• No more than 50 of each size are wanted.

• They need at least 20 poster size signs.

• No more than 75 signs are needed altogether.

• Letter-size signs cost $6.50 each, and poster-size signs cost $10.95 each.

The student council wants to minimize the cost of printing.

a) Create a model to represent this situation.

b) Suppose that there is an additional $15 fee to set up the printers. How would your model change?

18. Andrew has two summer jobs.

• He works no more than a total of 25 h a week. Both jobs allow him to have flexible hours but in whole hours only.

• At one job, Andrew works no less than 12 h and earns $9.00/h.

• At the other job, Andrew works no more than 20 h and earns $8.25/h.

What combination of numbers of hours will allow him to maximize his earnings? What can he expect to earn?

FOM11 Chapter 6 Review

Answer Section

MULTIPLE CHOICE

1. A

2. C

3. D

4. D

5. B

6. B

7. C

8. C

9. C

10. A

SHORT ANSWER

11. (

12. e.g., (10, 3) and (20, 5)

13. (

14. (

15. (0, 20)

16. Maximum: 132 legs

Minimum: 120 legs

PROBLEM

17. a) Let l represent the number of letter-size signs.

Let p represent the number of poster-size signs.

Let C represent the cost of printing.

l ∈ W, p ∈ W

l [pic] 50

20 [pic] p [pic] 50

l + p [pic] 75

Objective function to maximize:

C = 6.50l + 10.95p

b) The new objective function to maximize would be:

C = 15 + 6.50l + 10.95p

18. Let x represent the number of hours earning $9.00/h.

Let y represent the number of hours earning $8.25/h.

Let E represent the total earnings.

Restrictions:

x ∈ W, y ∈ W

Constraints:

x [pic] 12

y [pic] 20

x + y [pic] 25

Objective function to maximize:

E = 9.00x + 8.25y

Use technology to graph the lines and find the intersection points of the solution area.

The intersection points are (12, 0), (12, 13), and (25, 0).

The maximum will occur when x is maximized.

The maximum is at point (25, 0) and represents working all 25 h at $9.00/h.

R = 25(9.00) + 0(8.25)

R = 225

Her total earnings are $225.

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