Question 1

Graph the solution of the system of inequalities. ​   ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 2

Solve both the primal and dual problems with the simplex method. Use f to represent the dual objective function. Minimize   subject to

Accepted Answer

A) Primal:  
Dual:   
B) Primal:  
Dual:   
C) Primal:  
Dual:   
D) Primal:  
Dual:   
E) Primal:  
Dual:   
A) Primal:  
Dual:   
B) Primal:  
Dual:   
C) Primal:  
Dual:   
D) Primal:  
Dual:   
E) Primal:  
Dual:

Question 3

Solve the following linear programming problem. ​
Maximize   subject to
​   ​

Accepted Answer

A) 36 
B) 9 
C) 2 
D) 15 
E) 12 
A) 36 
B) 9 
C) 2 
D) 15 
E) 12

Question 4

Use the simplex method to maximize the function (if possible) subject to the given constraints. If there is no solution, indicate this; if multiple solutions exist, give one solution. ​
Maximize   subject to
​   ​

Accepted Answer

A) maximum: 56 at  
,   
B) maximum: 100 at  
,   
C) maximum: 84 at  
,   
D) maximum: 70 at  
,   
E) no solution 
A) maximum: 56 at  
,   
B) maximum: 100 at  
,   
C) maximum: 84 at  
,   
D) maximum: 70 at  
,   
E) no solution

Question 5

A political candidate wishes to use a combination of radio and TV advertisements in her campaign. Research has been shown that each 1-minute spot on TV reaches 0.6 million people and that each 1 -minute spot on radio reaches 0.4 million. The candidate feels she must reach 42 million people, and she must buy at least 54 minutes of advertisements. How many minutes of each medium should be used if TV costs $550 per minute, radio costs $110 per minute, and the candidate wishes to minimize costs? ​

Accepted Answer

A) Minimum cost is $15,950 using 10 minutes of TV time and 95 minutes of radio time. 
B) Minimum cost is $57,750 using 105 minutes of TV time and 0 minutes of radio time. 
C) Minimum cost is $57,750 using 0 minutes of TV time and 105 minutes of radio time. 
D) Minimum cost is $11,550 using 0 minutes of TV time and 105 minutes of radio time. 
E) Minimum cost is $11,550 using 105 minutes of TV time and 0 minutes of radio time. 
A) Minimum cost is $15,950 using 10 minutes of TV time and 95 minutes of radio time. 
B) Minimum cost is $57,750 using 105 minutes of TV time and 0 minutes of radio time. 
C) Minimum cost is $57,750 using 0 minutes of TV time and 105 minutes of radio time. 
D) Minimum cost is $11,550 using 0 minutes of TV time and 105 minutes of radio time. 
E) Minimum cost is $11,550 using 105 minutes of TV time and 0 minutes of radio time.

Question 6

Form the simplex matrix for the problem given below. Maximize   subject to

Accepted Answer

A)   
​ 
B)   
​ 
C)   
​ 
D)   
​ 
E)   
​ 
A)   
​ 
B)   
​ 
C)   
​ 
D)   
​ 
E)   
​

Question 7

A candidate wishes to use a combination of radio and television advertisements in her campaign. Assume that each 1-minute spot on television reaches to 0.0765 million people and that each 1-minute spot on radio reaches to 0.0045 million. The candidate feels she must reach at least 7.425 million people, and she must buy total of at least 90 minutes of advertisements. How many minutes of each medium should be used to minimize costs if television costs $450 per minute and radio costs $75 per minute? Round your answer to nearest integer if necessary.

Accepted Answer

A) Minimum cost is $43,676 with 0 minutes of TV time and 97 minutes of radio time. 
B) Minimum cost is $43,676 with 97 minutes of TV time and 0 minutes of radio time. 
C) Minimum cost is $9,154 with 5 minutes of TV time and 92 minutes of radio time. 
D) Minimum cost is $9,154 with 92 minutes of TV time and 5 minutes of radio time. 
E) Minimum cost is $7,654 with 0 minutes of TV time and 102 minutes of radio time. 
A) Minimum cost is $43,676 with 0 minutes of TV time and 97 minutes of radio time. 
B) Minimum cost is $43,676 with 97 minutes of TV time and 0 minutes of radio time. 
C) Minimum cost is $9,154 with 5 minutes of TV time and 92 minutes of radio time. 
D) Minimum cost is $9,154 with 92 minutes of TV time and 5 minutes of radio time. 
E) Minimum cost is $7,654 with 0 minutes of TV time and 102 minutes of radio time.

Question 8

Use Excel or another technology to solve the following optimization problem. Minimize   subject to

Accepted Answer

A) Minimum is 10,100 at   
B) Minimum is 9,100 at   
C) Minimum is 9,000 at   
D) Minimum is 10,000 at   
E) Minimum is 5,000 at   
A) Minimum is 10,100 at   
B) Minimum is 9,100 at   
C) Minimum is 9,000 at   
D) Minimum is 10,000 at   
E) Minimum is 5,000 at

Question 9

The graph of the feasible region is shown. Locate the corners of the feasible region in order to find the maximum of the given objective function (if it exists). ​   ​   ​

Accepted Answer

A) 240 
B) 360 
C) 70 
D) 100 
E) does not exist 
A) 240 
B) 360 
C) 70 
D) 100 
E) does not exist

Question 10

Three factories each dump waste water containing three different types of pollutants into a river. State regulations require the factories to treat their waste in order to reduce pollution levels. The table shows the possible percent reduction of each pollutant at each site and the cost per ton to process the waste. ​   ​
If the state requires a reduction of at least 65 tons per day of pollutant 1, at least 40 tons per day of pollutant 2, and at least 20 tons per day of pollutant 3, find the number of tons of waste that must be treated each day at each site so that the state's requirements are satisfied and the treatment costs are minimized. Find the minimum cost.
​
​

Accepted Answer

A) The minimum cost is $3,190 per day when Factory #2 and Factory #3 each process 110 tons and Factory #1 does not process any. 
B) The minimum cost is $2,920 per day when Factory #1 and Factory #2 each process 40 tons and Factory #3 does not process any. 
C) The minimum cost is $1,600 per day when Factory #3 process 200 tons, Factory #1 a and Factory #2 do not process any. 
D) The minimum cost is $1,540 per day when Factory #1 and Factory #2 each process 20 tons and Factory #3 produces 10 tons. 
E) The minimum cost is $2,490 per day when Factory #2 and Factory #3 each process 50 tons and Factory #1 produces 20 tons. 
A) The minimum cost is $3,190 per day when Factory #2 and Factory #3 each process 110 tons and Factory #1 does not process any. 
B) The minimum cost is $2,920 per day when Factory #1 and Factory #2 each process 40 tons and Factory #3 does not process any. 
C) The minimum cost is $1,600 per day when Factory #3 process 200 tons, Factory #1 a and Factory #2 do not process any. 
D) The minimum cost is $1,540 per day when Factory #1 and Factory #2 each process 20 tons and Factory #3 produces 10 tons. 
E) The minimum cost is $2,490 per day when Factory #2 and Factory #3 each process 50 tons and Factory #1 produces 20 tons.

Question 11

CDF Appliances has assembly plants in Atlanta and Fort Worth where they produce a variety of kitchen appliances, including a 12-cup coffee maker and a cappuccino machine. In each hour at the Atlanta plant, 160 of the 12-cup models and 200 of the cappuccino machines can be assembled and the hourly cost is $780. In each hour at the Fort Worth plant, 800 of the 12-cup models and 200 of the cappuccino machines can be assembled and the hourly cost is $2,120. CDF Appliances expects orders each week for at least 64,000 of the 12-cup models and at least 40,000 of the cappuccino machines. How many hours per week should each plant be operated in order to provide inventory for the orders at minimum cost? Find the minimum cost. ​

Accepted Answer

A) Minimum of $141,600 per week when Atlanta plant is operated for 100 hours and the Fort Worth plant is operated for 30 hours. 
B) Minimum of $178,400 per week when Atlanta plant is operated for 120 hours and the Fort Worth plant is operated for 40 hours. 
C) Minimum of $376,500 per week when Atlanta plant is operated for 75 hours and the Fort Worth plant is operated for 150 hours. 
D) Minimum of $343,000 per week when Atlanta plant is operated for 100 hours and the Fort Worth plant is operated for 125 hours. 
E) Minimum of $223,000 per week when Atlanta plant is operated for 150 hours and the Fort Worth plant is operated for 50 hours. 
A) Minimum of $141,600 per week when Atlanta plant is operated for 100 hours and the Fort Worth plant is operated for 30 hours. 
B) Minimum of $178,400 per week when Atlanta plant is operated for 120 hours and the Fort Worth plant is operated for 40 hours. 
C) Minimum of $376,500 per week when Atlanta plant is operated for 75 hours and the Fort Worth plant is operated for 150 hours. 
D) Minimum of $343,000 per week when Atlanta plant is operated for 100 hours and the Fort Worth plant is operated for 125 hours. 
E) Minimum of $223,000 per week when Atlanta plant is operated for 150 hours and the Fort Worth plant is operated for 50 hours.

Question 12

A farm co-op has 6,000 acres available to plant with corn and soybeans. Each acre of corn requires 9 gallons of fertilizer/herbicide and 0.75 hour of labor to harvest. Each acre of soybeans requires 3 gallons of fertilizer/herbicide and 1hour of labor to harvest. The co-op has available at most 40,501 gallons of fertilizer/herbicide and at most 5,250 hours of labor for harvesting. Find the maximum profit if the profits per acre are $75 for corn and $40 for soybeans. Round your answer to the nearest cent if necessary. ​

Accepted Answer

A) $210,000.00 
B) $393,750.00 
C) $371,255.83 
D) $180,004.44 
E) $318,744.17 
A) $210,000.00 
B) $393,750.00 
C) $371,255.83 
D) $180,004.44 
E) $318,744.17

Question 13

A primal maximization problem is given. Solve both the primal and dual problems with the simplex method. ​
Maximize   subject to
​   ​

Accepted Answer

A) ​Primal:  
Dual:  
​ 
B) ​Primal:  
Dual:  
​ 
C) ​Primal:  
Dual:  
​ 
D) ​Primal:  
Dual:  
​ 
E) ​​Primal:  
Dual:   
A) ​Primal:  
Dual:  
​ 
B) ​Primal:  
Dual:  
​ 
C) ​Primal:  
Dual:  
​ 
D) ​Primal:  
Dual:  
​ 
E) ​​Primal:  
Dual:

Question 14

Use the simplex method to find the optimal solution. ​
Maximize   subject to   ​

Accepted Answer

A)   
Maximum of   
B)   
Maximum of   
C)   
Maximum of   
D)   
Maximum of   
E)   
Maximum of   
A)   
Maximum of   
B)   
Maximum of   
C)   
Maximum of   
D)   
Maximum of   
E)   
Maximum of

Question 15

Use the simplex method to maximize the given function. Assume all variables are nonnegative. ​
Maximize   subject to
​   ​

Accepted Answer

A) maximum: 31 at  
,   
B) maximum: 41 at  
,   
C) maximum: 27 at  
,   
D) maximum: 32 at  
,   
E) maximum: 59 at  
,   
A) maximum: 31 at  
,   
B) maximum: 41 at  
,   
C) maximum: 27 at  
,   
D) maximum: 32 at  
,   
E) maximum: 59 at  
,

Question 16

State the following problem in a form for which the simplex matrix can be formed (that is, as a maximization problem with   constraints). ​   ​

Accepted Answer

A) ​  
​ 
B) ​  
​ 
C) ​  
​ 
D) ​  
​ 
E) ​   
A) ​  
​ 
B) ​  
​ 
C) ​  
​ 
D) ​  
​ 
E) ​

Question 17

Use the simplex method to find the optimal solution. Minimize   subject to

Accepted Answer

A) minimum: 37 at   
B) minimum: 15 at   
C) minimum: 25 at   
D) minimum: 12 at   
E) minimum: 22 at   
A) minimum: 37 at   
B) minimum: 15 at   
C) minimum: 25 at   
D) minimum: 12 at   
E) minimum: 22 at

Question 18

Suppose that in a hospital ward, the patients can be grouped into two general categories depending on their condition and the amount of solid foods they require in their diet. A combination of two diets is used for solid foods because they supply essential nutrients for recovery. The table below summarizes the patient groups and their minimum daily requirements. Let x represent number of servings of Diet A and y represent number of servings of Diet B. Write the inequalities that describe how many servings of each diet are needed to provide the nutritional requirements.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 19

A sausage company makes two different kinds of hot dogs, regular and all beef. Each pound of all-beef hot dogs requires 0.75 lb of beef and 0.2 lb of spices, and each pound of regular hot dogs requires 0.18 lb of beef, 0.3 lb of pork, and 0.2 lb of spices. Suppliers can deliver at most 1,020 lb of beef, at most 600 lb of pork, and at least 500 lb of spices. If the profit is $0.70 on each pound of all-beef hot dogs and $0.40 on each pound of regular hot dogs, how many pounds of each should be produced to obtain maximum profit? What is the maximum profit? Round your profit to the nearest cent, another answers - to the nearest whole number.
​

Accepted Answer

A) Maximum profit of $1150.00 with 2000 lbs of regular and 500 lbs of all beef. 
B) Maximum profit of $1416.00 with 2000 lbs of regular and 880 lbs of all beef. 
C) Maximum profit of $1752.00 with 880 lbs of regular and 2000 lbs of all beef. 
D) Maximum profit of $1300.00 with 1500 lbs of regular and 1000 lbs of all beef. 
E) Maximum profit of $1450.00 with 1000 lbs of regular and 1500 lbs of all beef. 
A) Maximum profit of $1150.00 with 2000 lbs of regular and 500 lbs of all beef. 
B) Maximum profit of $1416.00 with 2000 lbs of regular and 880 lbs of all beef. 
C) Maximum profit of $1752.00 with 880 lbs of regular and 2000 lbs of all beef. 
D) Maximum profit of $1300.00 with 1500 lbs of regular and 1000 lbs of all beef. 
E) Maximum profit of $1450.00 with 1000 lbs of regular and 1500 lbs of all beef.

Question 20

A chemical storage tank has a capacity of 200 tons. Currently, the tank contains 50 tons of a mixture that has 10% of a certain active chemical and 1.8% of other inert ingredients. The owners of the tank want to replenish the supply in the tank and will purchase some combination of two available mixes. Mix 1 contains 70% of the active chemical and 3% of the inert ingredients; its cost is $107 per ton. Mix 2 contains 30% of the active chemical and 1% of the inert ingredients; its cost is $46 per ton. The desired final mixture should have at least 40% of the active chemical and at most 2% of the inert ingredients. How many tons of each mix should be purchased to obtain the desired final mixture at minimum cost? Find the minimum cost. Note that at least 40% of the active chemical means

Accepted Answer

A) Minimum cost is $10,250 obtained by purchasing 70 tons of mix 1 and 60 tons of mix 2. 
B) Minimum cost is $8,720 obtained by purchasing 60 tons of mix 1 and 50 tons of mix 2. 
C) Minimum cost is $7,500 obtained by purchasing 40 tons of mix 1 and 70 tons of mix 2. 
D) Minimum cost is $9,180 obtained by purchasing 60 tons of mix 1 and 60 tons of mix 2. 
E) Minimum cost is $10,555 obtained by purchasing 50 tons of mix 1 and 60 tons of mix 2. 
A) Minimum cost is $10,250 obtained by purchasing 70 tons of mix 1 and 60 tons of mix 2. 
B) Minimum cost is $8,720 obtained by purchasing 60 tons of mix 1 and 50 tons of mix 2. 
C) Minimum cost is $7,500 obtained by purchasing 40 tons of mix 1 and 70 tons of mix 2. 
D) Minimum cost is $9,180 obtained by purchasing 60 tons of mix 1 and 60 tons of mix 2. 
E) Minimum cost is $10,555 obtained by purchasing 50 tons of mix 1 and 60 tons of mix 2.

Graph the solution of the system of inequalities.

Solve both the primal and dual problems with the simplex method. Use f to represent the dual objective function. Minimize subject to

Solve the following linear programming problem. Maximize subject to

Use the simplex method to maximize the function (if possible) subject to the given constraints. If there is no solution, indicate this; if multiple solutions exist, give one solution. Maximize subject to

Form the simplex matrix for the problem given below. Maximize subject to

Use Excel or another technology to solve the following optimization problem. Minimize subject to

The graph of the feasible region is shown. Locate the corners of the feasible region in order to find the maximum of the given objective function (if it exists).

A primal maximization problem is given. Solve both the primal and dual problems with the simplex method. Maximize subject to

Use the simplex method to find the optimal solution. Maximize subject to

Use the simplex method to maximize the given function. Assume all variables are nonnegative. Maximize subject to

State the following problem in a form for which the simplex matrix can be formed (that is, as a maximization problem with constraints).

Use the simplex method to find the optimal solution. Minimize subject to

Algebraic Concepts

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Functions of Two or More Variables

Filters

Exam 5: Inequalities and Linear Programming

Graph the solution of the system of inequalities. ​ ​