Deck 4: Linear Programming

Question 1

Sketch the graph of the equation x + 3y = 9.

Answer 1

Question 2

Find the point of intersection for the lines represented by the equations 2x + 4y = 12 and 3x + y = 13.

Answer 2

(4, 1)

Question 3

Write the constraint inequalities and the profit formula for this linear programming mixture problem: The Dig-the-Pig ham shop sells regular and special ham and cheese sandwiches. The regular is made of 5 oz of meat, 0.7 oz of cheese, and requires three minutes of preparation time. The special is made of 7 oz of meat, 0.6 oz of cheese, and requires 11 minutes of preparation time. The profit on a regular sandwich is 10 cents while the profit on a special sandwich is 40 cents. If Dig-the-Pig has 350 oz of meat, 42 oz of cheese, and 330 minutes of preparation time available each lunch period, how many of each type of sandwich should they try to sell to maximize profit?

Answer 3

x is the number of regular sandwiches made and y is the number of special sandwiches made.
Constraint inequalities: 5x + 7y  3500.7x + 0.6y  42
3x + 11y  330
x  0
y  0
Profit formula: P = 10x + 40y

Question 4

Find the point of intersection for the lines represented by the equations 3x + 2y = 16 and 4x + 3y = 23.

Question 5

With the given constraints for the following linear programming mixture problem, graph the feasible region. 2 x + 3y  1800
x  0
y  0

Question 6

Find the point of intersection for the lines represented by the equations 2x + 7y = 61 and 3x + 4y = 46.

Question 7

Describe a mixture problem.

Question 8

With the given constraints for the following linear programming mixture problem, graph the feasible region. x + 3y  23
3x + 7y  50
x  2
y  4

Question 9

Find the constraint inequalities and the profit formula for this linear programming mixture problem: The Acme Construction Company builds two types of houses. Plan A requires 200 man-hours for rough construction and 70 man-hours for finish work. Plan B requires 300 man-hours for rough construction and 50 man-hours for finish work. Acme has carpenters available to provide up to 900 hours of rough construction per month and 260 hours of finish work per month. If Acme clears $7000 profit on a Plan A house and $8000 profit on a Plan B house, how should they schedule production to maximize profit?

Question 10

With the given constraints for the following linear programming mixture problem, graph the feasible region. x + 6y  1100
4x + y  2100
x  0
y  0

Question 11

Sketch the graph of the inequality 2x + 4y  12.

Question 12

Find the point of intersection for the lines represented by the equations 3x + 2y = 14 and 4x + 5y = 28.

Question 13

Sketch the graph of the inequality 5x + y  15.

Question 14

Sketch the graph of the inequality 4x + 6y  12.

Question 15

Find the constraint inequalities and the profit formula for this linear programming mixture problem: Toni has a small business producing dried floral wreaths and table arrangements. Each wreath takes seven hours to produce, uses 12 stems of flowers, and earns a profit of $23. Each table arrangement takes five hours to produce, uses 20 stems of flowers, and earns a profit of $26. Toni can work no more than 30 hours per week and has a steady supply of 100 stems of dried flowers per week. How should Toni schedule production to optimize profit?

Question 16

Write the constraint inequalities and the profit formula for this linear programming mixture problem: Amazin' Raisin Baking Co. makes both raisin cake and raisin pie. A batch of raisin cakes requires 5 lbs of flour, 2 lbs of sugar, and 1 lb of raisins. A batch of raisin pies requires 2 lbs of flour, 3 lbs of sugar, and 4 lbs of raisins. There are 165 lbs of flour, 110 lbs of sugar, and 120 lbs of raisins available each week. Standing orders require at least five batches of raisin cakes and eight batches of raisin pies per week. If profit on a batch of raisin cakes is $35 and profit on a batch of raisin pies is $40, how many batches of each should be made per week to maximize profit?

Question 17

With the given constraints for the following linear programming mixture problem, graph the feasible region. 2 x + 3y  180
5 x + 2y  230
x  0
y  0

Question 18

What is the goal of a linear programming?

Question 19

With the given constraints for the following linear programming mixture problem, graph the feasible region. x + 6y  110
x  0
y  0

Question 20

Solve this linear programming mixture problem: Kim and Lynn produce pottery vases and bowls. A vase requires 25 oz of clay and 5 oz of glaze. A bowl requires 20 oz of clay and 10 oz of glaze. There are 500 oz of clay available and 160 oz of glaze available. The profit on one vase is $5 and the profit on one bowl is $3. How should Kim and Lynn schedule production to optimize profit?

Question 21

What type of solution does the Stepping Stone Method produce for the transportation problem?

Question 22

Solve this linear programming mixture problem: A small stereo manufacturer makes a receiver and a CD player. Each receiver takes eight hours to assemble, one hour to test and ship, and earns a profit of $30. Each CD player takes 15 hours to assemble, two hours to test and ship, and earns a profit of $50. There are 160 hours available in the assembly department and 26 hours available in the testing and shipping department. What should the production schedule be to maximize profit?

Question 23

In the optimal solution, how many units are shipped from supplier I to customer 2?

Question 24

Explain what the real world implications are if the optimal production policy for a linear programming mixture problem is represented by a point on the x-axis of the Cartesian plane.

Question 25

What is a tableau of a transportation problem?

Question 26

What is the indicator value of cell (III, 1)?

Question 27

Explain why the feasible region for a linear programming mixture problem must be in the first quadrant of the Cartesian plane.

Question 28

Name two alternatives to the graphical approach for solving linear programming problems.

Question 29

Find the point of intersection of the lines whose equations are 6x + 5y = 80 and 2x + y = 20.

Question 30

What is the indicator value of cell (II, 2)?

Question 31

What is the goal of a transportation problem?

Question 32

Find the maximum value of P, where P = 3x + 4y subject to the constraints:
x  0; y  0 ; x + 2y  8; x + y  5

Question 33

Find the maximum value of P, where P = 100x + 10y subject to the constraints:
x  0; y  0 ; 2x + 5y  20; 2x + y  12

Question 34

Fill in the blank: Given a transportation problem, a collection of circled cells is optimal if the value associated with each of the empty cells is positive.

Question 35

What is the cost of the optimal solution to this shipping problem?

Question 36

A solution for the transportation problem is optimal when the empty cells have what property?

Question 37

What type of solution does the Northwest Corner Rule produce for the transportation problem?

Question 38

What is the cost of the solution shown in the tableau above?

Question 39

Describe the shape of the feasible region for linear programming mixture problems with two products.

Question 40

Explain why a linear programming mixture problem might have minimum constraints other than zero.

Question 41

Find the point of intersection of the lines with equations 4x + 2y = 12 and 3x + 9y = 39.

A) (5, -4)
B) (10, 1)
C) (1, 4)
D) (2, 2)

Question 42

Find the graph of the inequality 3x + 4y  12.

Question 43

Find the graph of the inequality 4x + 5y  40.

Question 44

Find the graph of the equation 4x + 6y = 18.

Question 45

Write a resource constraint for this situation: Producing a plastic ruler (x) requires 10 grams of plastic and producing a pencil box (y) requires 30 grams of plastic. There are 2000 grams of plastic available.

A) 200x + (2000/30)y  2000
B) 30x + 10y  2000
C) 10x + 30y  2000
D) x + y  2000

Question 46

Find the graph of the inequality 3x + 7y  21.

Question 47

Write the constraint inequalities for this situation: A cheeseburger (x) requires 5 oz of meat and 0.7 oz of cheese and a superburger (y) requires 7 oz of meat and 0.6 oz of cheese. The burger stand has 350 oz of meat and 42 oz of cheese available. The profit on a cheeseburger is 10 cents and the profit on a superburger is 40 cents.

A) 5x + 7y  350, 0.7 x + 0.6y  42, x  0, y  0
B) 5x + 0.7y  10, 7 x + 0.6y  40, x  0, y  0
C) 5x + 7y  10, 0.7 x + 0.6y  40, x  0, y  0
D) 70x + 50y  350, 60 x + 70y  42, x  0, y  0

Question 48

Find the point of intersection of the lines with equations 2x + 5y = 6 and 3x + 2y = 9.

A) (3, 0)
B) (2, 1)
C) (-3, 0)
D) (1, 2)

Question 49

Find the graph of the inequality 6x + 4y  48.

Question 50

Write the constraint inequalities for this situation: Kim and Lynn produce pottery vases and bowls. A vase requires 35 oz of clay and 5 oz of glaze. A bowl requires 20 oz of clay and 10 oz of glaze. There are 500 oz of clay available and 200 oz of glaze available. The profit on one vase is $5 and the profit on one bowl is $4.

A) 35x + 5y  5, 20 x + 10y  4, x  0, y  0
B) 35x + 5y  500, 20 x + 10y  200, x  0, y  0
C) 35x + 20y  500, 5 x + 10y  200, x  0, y  0
D) 35x + 20y  $5, 5 x + 10y  $4, x  0, y  0

Question 51

For a linear programming problem, the set/region is the collection of all physically possible solution choices that can be made.

A) population
B) sample
C) feasible
D) characteristic

Question 52

Find the point of intersection of the lines with equations x + 4y = 11 and 2x + 5y = 16.

A) (2, 3)
B) (3, 2)
C) (6, 0)
D) (-2, 3)

Question 53

Find the graph of the equation 5x + 2y = 15.

Question 54

Graph the constraint inequalities for a linear programming problem shown below. Which feasible region shown is correct?

Question 55

Which of the following descriptions is NOT a correct common feature of mixture problem?

A) Definite resources are available in limited, known quantities for the period in question.
B) Definite products can be made by combining, or mixing the resources.
C) Each product's profit is contingent upon time.
D) The objective in a mixture problem is to find how much of each product to make so as to maximize the profit without exceeding any of the resource limitations.

Question 56

Graph the constraint inequalities for a linear programming problem shown below. Which feasible region shown is correct?

Question 57

Graph the constraint inequalities for a linear programming problem shown below. Which feasible region shown is correct?

Question 58

Find the point of intersection of the lines with equations 3x + 2y = 21 and 2x + 1y = 13.

A) (5, 3)
B) (29, 45)
C) (8, -3)
D) (3, 5)

Question 59

Graph the constraint inequalities for a linear programming problem shown below. Which feasible region shown is correct?

Question 60

Write a resource constraint for this situation: A lawn service company has 40 hours of worker time available. Mowing a lawn (x) takes three hours and trimming (y) takes two hours. The profit from mowing is $15 and the profit from trimming is $10.

A) 3x + 2y  40
B) (40/3)x + 10y  40
C) 15x + 10y  40
D) 5x + 5y  40

Question 61

Write a profit formula for this mixture problem: A small stereo manufacturer makes a receiver (x) and a CD player (y). Each receiver takes eight hours to assemble, one hour to test and ship, and earns a profit of $30. Each CD player takes 15 hours to assemble, two hours to test and ship, and earns a profit of $50. There are 160 hours available in the assembly department and 22 hours available in the testing and shipping department.

A) P = 8x + 1y
B) P = 160x + 22y
C) P = 15x + 2y
D) P = 30x + 50y

Question 62

The graph of the feasible region for a mixture problem is shown below. Find the point that maximizes the profit function P = 2x + y.

A) (0, 2)
B) (2, 4)
C) (4, 1)
D) (3, 0)

Question 63

Write the resource constraints for this situation: A company manufactures patio chairs (x) and rockers (y). Each piece is made of wood, plastic, and aluminum. A chair requires one unit of wood, one unit of plastic, and two units of aluminum. A rocker requires one unit of wood, two units of plastic, and five units of aluminum. The company's profit on a chair is $7 and on a rocker is $12. The company has available 400 units of wood, 500 units of plastic, and 1450 units of aluminum.

A) 1x + 1y + 2z  7, 1x + 2y + 5z  12, x  0, y  0, z  0
B) 1x + 1y  400, 1 x + 2y  500, 2x + 5y  1450, x  0, y  0
C) 400x + 500y + 1450z  0, 1x + 1y + 2z  7, 1x+ 2y + 5z  12, x  0, y  0, z  0
D) 7x + 12y  400, 2 x + 5y  1450, x  0, y  0

Question 64

The graph of the feasible region for a mixture problem is shown below. Find the point that maximizes the profit function P = 2x + 3y.

A) (0, 5)
B) (3, 4)
C) (7, 2)
D) (9, 0)

Question 65

Write a profit formula for this mixture problem: Kim and Lynn produce pottery vases (x) and bowls (y). A vase requires 35 oz of clay and 5 oz of glaze. A bowl requires 20 oz of clay and 10 oz of glaze. There are 500 oz of clay available and 200 oz of glaze available. The profit on one vase is $5 and the profit on one bowl is $4.

A) P = 500x + 200y
B) P = 35x + 20y
C) P = 5x + 4y
D) P = 5x + 10y

Question 66

Graph the feasible region identified by the inequalities:

Question 67

Given below is the sketch of the feasible region in a linear programming problem. Which point is NOT in the feasible region?

A) (0, 4)
B) (4, 0)
C) (6, 0)
D) (1, 2)

Question 68

Write the resource constraints for this situation: Kim and Lynn produce tables (x) and chairs (y). Each piece is assembled, sanded, and stained. A table requires two hours to assemble, three hours to sand, and three hours to stain. A chair requires four hours to assemble, two hours to sand, and three hours to stain. The profit earned on each table is $20 and on each chair is $12. Together Kim and Lynn spend at most 16 hours assembling, 10 hours sanding, and 13 hours staining.

A) 2x + 4y  16, 3 x + 2y  10, 3x + 3y  13, x  0, y  0
B) 2x + 3y + 3z  20, 4x + 2y + 3z  12, x  0, y  0, z  0
C) 16x + 10y + 13z  0, 2x + 3y + 3z  20, 4x + 2y + 3z  12, x  0, y  0, z  0
D) 8x + 4y  16, (10/3) x + 5y  10, (13/3)x + (13/3)y  13, x  0, y  0

Question 69

Graph the feasible region identified by the inequalities:

Question 70

Given below is the sketch of the feasible region in a linear programming problem. Which point is NOT in the feasible region?

A) (0, 8)
B) (12, 0)
C) (6, 4)
D) (2, 2)

Question 71

Graph the feasible region identified by the inequalities:

Question 72

Write a profit formula for this mixture problem: Kim and Lynn produce tables (x) and chairs (y). Each piece is assembled, sanded, and stained. A table requires two hours to assemble, three hours to sand, and three hours to stain. A chair requires four hours to assemble, two hours to sand, and three hours to stain. The profit earned on each table is $20 and on each chair is $12. Together Kim and Lynn spend at most 16 hours assembling, 10 hours sanding, and 13 hours staining.

A) P = 20x + 12y
B) P = 2x + 3y + 3z
C) P = 16x + 10y + 13z
D) P = 8x + 9y

Question 73

Given below is the sketch of the feasible region in a linear programming problem. Which point is NOT in the feasible region?

A) (6, 4)
B) (0, 10)
C) (2, 6)
D) (0, 8)

Question 74

Graph the feasible region identified by the inequalities:

Question 75

The graph of the feasible region for a mixture problem is shown below. Find the point that maximizes the profit function P = 3x + 6y.

A) (0, 4)
B) (3, 3)
C) (5, 1)
D) (6, 0)

Question 76

Write the resource constraints for this situation: A small stereo manufacturer makes a receiver and a CD player. Each receiver (x) takes eight hours to assemble and eight hours to test and ship. Each CD player (y) takes 15 hours to assemble and two hours to test and ship. The profit on each receiver is $30 and the profit on each CD player is $50. There are 160 hours available in the assembly department and 22 hours available in the testing and shipping department.

A) 8x + 1y  30, 15x + 2y  50, x  0, y  0
B) 8x + 1y  160, 15x + 2y  22, x  0, y  0
C) 8x + 15y  30, 8x + 2y  50, x  0, y  0
D) 8x + 15y  160, 8x + 2y  22, x  0, y  0

Question 77

Given below is the sketch of the feasible region in a linear programming problem. Which point is NOT in the feasible region?

A) (0, 32)
B) (0, 24)
C) (8, 16)
D) (20, 12)

Question 78

Write a profit formula for this mixture problem: A company manufactures patio chairs (x) and rockers (y). Each piece is made of wood, plastic, and aluminum. A chair requires one unit of wood, one unit of plastic, and two units of aluminum. A rocker requires one unit of wood, two units of plastic, and five units of aluminum. The company's profit on a chair is $7 and on a rocker is $12. The company has available 400 units of wood, 500 units of plastic, and 1450 units of aluminum.

A) P = 400x + 500y + 1450z
B) P = 4x + 8y
C) P = 7x + 12y
D) P = 1x + 2y + 5z

Question 79

The graph of the feasible region for a mixture problem is shown below. Find the point that maximizes the profit function P = x + 4y.

A) (0, 9)
B) (6, 7)
C) (7, 3)
D) (6, 0)

Question 80

Given below is the sketch of the feasible region in a linear programming problem. Which point is NOT in the feasible region?

A) (0, 6)
B) (4, 0)
C) (4, 2)
D) (6, 0)