Deck 5: Linear Inequalities and Linear Programming

Question 1

Use graphical methods to solve the linear programming problem.
Minimize z = 2x + 4y subject to:
x + 2y ≥ 10
3x + y ≥ 10
x ≥ 0
Y ≥ 0

A)Minimum of 0 when x = 0 and y = 0
B)Minimum of 20 when x = 10 and y = 0
C)Minimum of 20 when x = 2 and y = 4
D)Minimum of 20 when x = 2 and y = 4,as well as when x = 10 and y = 0,and all points in between

Answer 1

Minimum of 20 when x = 2 and y = 4,as well as when x = 10 and y = 0,and all points in between

Question 2

Use graphical methods to solve the linear programming problem.
Minimize z = 4x + 5y subject to:
2x - 4y ≤ 10
2x + y ≥ 15
x ≥ 0
Y ≥ 0

A)Minimum of 33 when x = 7 and y = 1
B)Minimum of 39 when x = 1 and y = 7
C)Minimum of 20 when x = 5 and y = 0
D)Minimum of 75 when x = 0 and y = 15

Answer 2

Minimum of 33 when x = 7 and y = 1

Question 3

Graph the constant-profit lines through (3, 2) and (5, 3). Use a straightedge to identify the corner point(s) where the maximum profit occurs for the given objective function.
P = x + y

A)Max P = 8 at x = 5 and y = 3
B)Max P = 5 at x = 3 and y = 2
C)Max P = 9 at x = 9 and y = 0,at x = 8 and y = 4,and at every point on the line segment joining the preceding two points.
D)Max P = 12 at x = 0 and y = 12,at x = 8 and y = 4,and at every point on the line segment joining the preceding two points.

Answer 3

Max P = 12 at x = 0 and y = 12,at x = 8 and y = 4,and at every point on the line segment joining the preceding two points.

Question 4

Solve the problem.
Jim has gotten scores of 78 and 69 on his first two tests.What score must he get on his third test to keep an average of 70 or greater?

A)At least 63
B)At least 62
C)At least 72.3
D)At least 73.5

Question 5

Solve the problem.
A salesperson has two job offers.Company A offers a weekly salary of $180 plus commission of 6% of sales. Company B offers a weekly salary of $360 plus commission of 3% of sales.What is the amount of sales above
Which Company A's offer is the better of the two?

A)$12,000
B)$3000
C)$6000
D)$6100

Question 6

Provide an appropriate response.
Solve the following linear programming problem by determining the feasible region on the graph below and testing the corner points:
Minimize C = x₁ + 6x₂ subject to
3x₁ + 4x₂ ≥ 36
2x₁ + x₂ ≤ 14
x₁, x₂ ≥ 0

x₁ is shown on the x-axis and x₂ on the y-axis.

Question 7

Provide an appropriate response.
The corner points for the bounded feasible region determined by the system of inequalities:
2x₁ + 5x₂ ≤ 20
x₁ + x₂ ≤ 7
x₁,x₂ ≥ 0
are O = (0,0),A = (0,4),B = (5,2)and C = (7,0).Find the optimal solution for the objective profit function: P(x)= 3x₁ + 7x₂

Question 8

Provide an appropriate response.
The corner points for the bounded feasible region determined by the system of inequalities:
5x₁ + 2x₂ ≤ 40
x₁ + 3x₂ ≤ 21
x₁,x₂ ≥ 0
are O = (0,0),A = (0,7),B = (6,5)and C = (8,0).Find the optimal solution for the objective profit function: P = 5x₁ + 5x₂

Question 9

Define the variable(s) and translate the sentence into an inequality.
Enrollment is below 8000 students.

A)Let e = student enrollment; e < 8000
B)Let e = student enrollment; e ≤ 8000
C)Let e = student enrollment; e > 8000
D)Let e = student enrollment; e ≥ 8000

Question 10

Provide an appropriate response.
Find the coordinates of the corner points of the solution region for:
3x + 2y ≥ 54
4x + 5y ≤ 100
x ≥ 0
y ≥ 0

Question 11

Graph the constant-profit lines through (3, 2) and (5, 3). Use a straightedge to identify the corner point(s) where the maximum profit occurs for the given objective function.
P = 5x + y

A)Max P = 45 at x = 9 and y = 0,at x = 8 and y = 4,and at every point on the line segment joining the preceding two points.
B)Max P = 44 at x = 8 and y = 4,at x = 0 and y = 12,and at every point on the line segment joining the preceding two points.
C)Max P = 44 at x = 8 and y = 4
D)Max P = 45 at x = 9 and y = 0

Question 12

Solve the problem.
Company A rents copiers for a monthly charge of $300 plus 10 cents per copy.Company B rents copiers for a monthly charge of $600 plus 5 cents per copy.What is the number of copies above which Company A's charges are the higher of the two?

A)12,000 copies
B)3000 copies
C)6000 copies
D)6100 copies

Question 13

Define the variable(s) and translate the sentence into an inequality.
Sales of wheat bread are at least $2000 greater than sales of white bread.

A)Let e = wheat bread sales; let i = white bread sales; w ≥ i + $2000
B)Let e = wheat bread sales; let i = white bread sales; w + i ≥ $2000
C)Let e = wheat bread sales; let i = white bread sales; i ≥ w + $2000
D)Let e = wheat bread sales; let i = white bread sales; w ≤ i + $2000

Question 14

Graph the solution set of the system of linear inequalities and indicate whether the solution region is bounded or unbounded.
y ≤ -3x - 3
y ≥ x + 6

A)
B)
C)
D)

Question 15

Provide an appropriate response.
Refer to the following system of linear inequalities associated with a linear programming problem:
Maximize P = 3x₁ + 7x₂ subject to
5x₁ + x₂ ≤ 28
2x₁ + x₂ ≤ 13
x₁ + x₂ ≤ 0
x₁, x₂ ≥ 0
(A)Determine the number of slack variable that must be introduced to form a system of problem constraint equations.
(B)Determine the number of basic variables associated with this system.

Question 16

Use graphical methods to solve the linear programming problem.
Maximize z = 8x + 12y subject to:
40x + 80y ≤ 560
6x + 8y ≤ 72
x ≥ 0
Y ≥ 0

A)Maximum of 100 when x = 8 and y = 3
B)Maximum of 96 when x = 9 and y = 2
C)Maximum of 120 when x = 3 and y = 8
D)Maximum of 92 when x = 4 and y = 5

Question 17

Use graphical methods to solve the linear programming problem.
A math camp wants to hire counselors and aides to fill its staffing needs at minimum cost.The average monthly salary of a counselor is $2400 and the average monthly salary of an aide is $1100.The camp can accommodate up to 45 staff members and needs at least 30 to run properly.They must have at least 10 aides,and may have up to 3 aides for every 2 counselors.How many counselors and how many aides should the camp hire to minimize cost?

A)27 counselors and 18 aides
B)18 counselors and 12 aides
C)35 counselors and 10 aides
D)12 counselors and 18 aides

Question 18

Use graphical methods to solve the linear programming problem.
Maximize z = 6x + 7y subject to:
2x + 3y ≤ 12
2x + y ≤ 8
x ≥ 0
Y ≥ 0

A)Maximum of 32 when x = 3 and y = 2
B)Maximum of 24 when x = 4 and y = 0
C)Maximum of 52 when x = 4 and y = 4
D)Maximum of 32 when x = 2 and y = 3

Question 19

Provide an appropriate response.
Using a graphing calculator as needed,maximize P = 524x₁ + 479x₂ subject to
265x1 + 320x₂ ≤ 3,390
350x₁ + 345x₂ ≤ 3,795
400x₁ + 316x₂ ≤ 4,140
x1,x2 ≥ 0
Give the answer to two decimal places.

Question 20

Graph the solution set of the system of linear inequalities and indicate whether the solution region is bounded or unbounded.
y ≤ -2x + 2
y ≥ x - 7

A)
B)
C)
D)

Question 21

Solve the problem.
Formulate the following problem as a linear programming problem (DO NOT SOLVE):A steel company produces two types of machine dies,part A and part B.Part A requires 6 hours of casting time and 4 hours of firing time.Part B requires 8 hours of casting time and 3 hours of firing time.The maximum number of hours per week available for casting and firing are 85 and 70,respectively.The company makes a $2.00 profit on each part A that it produces,and a $6.00 profit on each part B that it produces.How many of each type should the company produce each week in order to maximize its profit? (Let x1 equal the number of A parts and x2 equal the number of B parts produced each week.)

Question 22

Use graphical methods to solve the linear programming problem.
Suppose an horse feed to be mixed from soybean meal and oats must contain at least 100 lb of protein,20 lb of fat,and 9 lb of mineral ash.Each 100-lb sack of soybean meal costs $20 and contains 50 lb of protein,10 lb of fat,and 8 lb of mineral ash.Each 100-lb sack of oats costs $10 and contains 20 lb of protein,5 lb of fat,and 1 lb of mineral ash.How many sacks of each should be used to satisfy the minimum requirements at minimum cost?

A) 2 sacks of soybeans and 0 sacks of oats
B) sacks of soybeans and sacks of oats
C) sacks of soybeans and sacks of oats
D) 0 sacks of soybeans and 2 sacks of oats

Question 23

Suppose an horse feed to be mixed from soybean meal and oats must contain at least 200 lb of protein and 40 lb of fat.Each sack of soybean meal costs $20 and contains 60 lb of protein and 10 lb of fat.Each sack of oats costs
$10 and contains 20 lb of protein and 5 lb of fat.How many sacks of each should be used to satisfy the minimum requirements at minimum cost?

Question 24

The Southern States Ring Company designs and sells two types of rings: the brass and the aluminum.They can produce up to 24 rings each day using up to 60 total man-hours of labor per day.It takes 3 man-hours to make one brass ring and 2 man-hours to make one aluminum ring.How many of each type of ring should be made daily to maximize the company's profit,if the profit on a brass ring is $40 and on an aluminum ring is $35?

Question 25

Solve the problem.
Formulate the following problem as a linear programming problem (DO NOT SOLVE):A dietitian can purchase an ounce of chicken for $0.25 and an ounce of potatoes for $0.02.Each ounce of chicken contains 13 units of protein and 24 units of carbohydrates.Each ounce of potatoes contains 5 units of protein and 35 units of carbohydrates.The minimum daily requirements for the patients under the dietitian's care are 45 units of protein and 58 units of carbohydrates.How many ounces of each type of food should the dietitian purchase for each patient so as to minimize costs and at the same time insure the minimum daily requirements of protein
and carbohydrates? (Let x₁ equal the number of ounces of chicken and x₂ the number of ounces of potatoes purchased per patient.)

Question 26

Use graphical methods to solve the linear programming problem.
The Old-World Class Ring Company designs and sells two types of rings: the BRASS and the GOLD.They can produce up to 24 rings each day using up to 60 total man-hours of labor.It takes 3 man-hours to make one BRASS ring and 2 man-hours to make one GOLD ring.How many of each type of ring should be made daily to maximize the company's profit,if the profit on a BRASS ring is $40 and on an GOLD ring is $30?

A)14 BRASS and 10 GOLD
B)12 BRASS and 12 GOLD
C)14 BRASS and 14 GOLD
D)10 BRASS and 14 GOLD

Question 27

Solve the problem.
A vineyard produces two special wines a white,and a red.A bottle of the white wine requires 14 pounds of grapes and 1 hour of processing time.A bottle of red wine requires 25 pounds of grapes and 2 hours of processing time.The vineyard has on hand 2,198 pounds of grapes and can allot 160 hours of processing time to the production of these wines.A bottle of the white wine sells for $11.00,while a bottle of the red wine sells for
$20.00.How many bottles of each type should the vineyard produce in order to maximize gross sales? (Solve using the geometric method.)

Question 28

A math camp wants to hire counselors and aides to fill its staffing needs at minimum cost.The monthly salary of a counselor is $2400 and the monthly salary of an aide is $1100.The camp can accommodate up to 45 staff members and needs at least 30 to run properly.They must have at least 10 aides,and at most twice as many aides as counselors.How many counselors and how many aides should the camp hire to minimize cost?