Question 1

Provide an appropriate response.
-Describe the feasible region of $x+y \geq 9$ and $x+y \leq-6$.

Accepted Answer

A)  Two bounded regions 
B)  A bounded region 
C)  An empty region 
D)  An unbounded region 
A)  Two bounded regions 
B)  A bounded region 
C)  An empty region 
D)  An unbounded region

Question 2

Use duality to solve the problem.
-Minimize $w=5 y 1+3 y_{2}$
Subject to: $2 \mathrm{y}_{1}+3 \mathrm{y}_{2} \geq 9$
$2 \mathrm{y}_{1}+\mathrm{y}_{2} \geq 11$
$\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0$

Accepted Answer

A)  33 
B)  27 
C)  25 
D)  27.5 
A)  33 
B)  27 
C)  25 
D)  27.5

Question 3

Each day Larry needs at least 10 units of vitamin A, 12 units of vitamin B, and 20 units of vitamin C. Pill \#1 contains 4 units of $A$ and 3 of B. Pill \#2 contains 1 unit of A, 2 of B, and 4 of C. Pill \#3 contains 10 units of A, 1 of B, and 5 of $C$.
-Pill #1 costs 2 cents, pill #2 costs 5 cents, and pill #3 costs 8 cents. Larry wants to minimize cost. What is the constraint inequality for vitamin $\mathrm{B}$ ?

Accepted Answer

A)  $3 \mathrm{P}_{1}+2 \mathrm{P}_{2}+\mathrm{P}_{3} \geq 12$ 
B)  $3 \mathrm{P}_{1}+2 \mathrm{P}_{2}+\mathrm{P}_{3} \leq 5$ 
C)  $2 \mathrm{P}_{2}+5 \mathrm{P}_{3}+8 \mathrm{P}_{3} \geq 12$ 
D)  $4 P_{1}+P_{2}+10 P_{3} \geq 10$ 
A)  $3 \mathrm{P}_{1}+2 \mathrm{P}_{2}+\mathrm{P}_{3} \geq 12$ 
B)  $3 \mathrm{P}_{1}+2 \mathrm{P}_{2}+\mathrm{P}_{3} \leq 5$ 
C)  $2 \mathrm{P}_{2}+5 \mathrm{P}_{3}+8 \mathrm{P}_{3} \geq 12$ 
D)  $4 P_{1}+P_{2}+10 P_{3} \geq 10$

Question 4

Introduce slack variables as necessary and write the initial simplex tableau for the problem.
-Maximize
$$z=4 x_{1}+2 x_{2}$$
Subject to:
$$2 x_{1}+x_{2} \leq 25$$
$$\begin{array}{r}
3 x_{1}+5 x_{2} \leq 74 \
X_{1} \geq 0, x_{2} \geq 0
\end{array}$$

Accepted Answer

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

Question 5

Use the indicated entry as the pivot and perform the pivoting once.
-

Accepted Answer

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

Question 6

Convert the objective function into a maximization function.
-Minimize $w=y_{1}+y_{2}+5 y_{3}$
Subject to: $\mathrm{y}_{1}+\mathrm{y}_{3} \geq 28$
$\mathrm{y}_{1}+2 \mathrm{y}_{2}+3 \mathrm{y}_{3} \geq 59$
$2 \mathrm{y}_{1}+3 \mathrm{y}_{2}+2 \mathrm{y}_{3} \geq 85$
$\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0, \mathrm{y}_{3} \geq 0$

Accepted Answer

A)  Maximize $z=-y_{1}-y_{2}-5 y_{3}$ 
B)  Maximize $z=-y_{1}-y_{3} \leq 28$ 
C)  Maximize $z=y_{1}+y_{2}+5 y_{3}-y_{4}$ 
D)  Maximize $z=y_{1}-2 y_{2}-3 y_{3} \leq 59$ 
A)  Maximize $z=-y_{1}-y_{2}-5 y_{3}$ 
B)  Maximize $z=-y_{1}-y_{3} \leq 28$ 
C)  Maximize $z=y_{1}+y_{2}+5 y_{3}-y_{4}$ 
D)  Maximize $z=y_{1}-2 y_{2}-3 y_{3} \leq 59$

Question 7

A manufacturer of wooden chairs and tables must decide in advance how many of each item will be made in a givenweek. Use the table to find the system of inequalities that describes the manufacturer's weekly production.
-Use $x$ for the number of chairs and $y$ for the number of tables made per week. The number of work-hours available for construction and finishing is fixed.

Accepted Answer

A)  $2 x+4 y+48 \leq 0$
$2 x+3 y+42 \leq 0$ 
B)  $2 x+4 y \geq 48$
$2 x+3 y \geq 42$
$x \geq 0$
$\mathrm{y} \geq 0$ 
C)  $2 x+4 y \leq 48$
$2 x+3 y \leq 42$ 
D)  $2 x+4 y \leq 48$
$2 x+3 y \leq 42$
$x \geq 0$
$y \geq 0$ 
A)  $2 x+4 y+48 \leq 0$
$2 x+3 y+42 \leq 0$ 
B)  $2 x+4 y \geq 48$
$2 x+3 y \geq 42$
$x \geq 0$
$\mathrm{y} \geq 0$ 
C)  $2 x+4 y \leq 48$
$2 x+3 y \leq 42$ 
D)  $2 x+4 y \leq 48$
$2 x+3 y \leq 42$
$x \geq 0$
$y \geq 0$

Question 8

Provide an appropriate response.
-No unique optimum solution found from a simplex tableau corresponds to

Accepted Answer

An unbound

Question 9

Find the value(s) of the function on the given feasible region.
-Find the maximum and minimum of $z=8 x-9 y$.

Accepted Answer

A)  $-54,0$ 
B)  $40,-54$ 
C)  $-35,-54$ 
D)  40,0 
A)  $-54,0$ 
B)  $40,-54$ 
C)  $-35,-54$ 
D)  40,0

Question 10

Graph the linear inequality.
-$x+2 y \geq-1$

Accepted Answer

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

Question 11

Write the word or phrase that best completes each statement or answers thequestion.
-Explain how you decide which half-plane to shade when you are graphing an inequality.

Accepted Answer

Use either method:
1) Solve for @#LAT-DLM&; shade

Question 12

Graph the linear inequality.
-$2 x+5 y \leq 10$

Accepted Answer

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

Question 13

Provide an appropriate response.
-If the inequalities $x \geq 0$ and $y \geq 0$ are included in a system, the feasibility region is restricted to the axes and which quadrant?

Accepted Answer

A)  Fourth 
B)  First 
C)  Second 
D)  It is not restricted. 
A)  Fourth 
B)  First 
C)  Second 
D)  It is not restricted.

Question 14

A manufacturing company wants to maximize profits on products $A, B$, and $C$. The profit margin is $\$ 3$ for $A, \$ 6$ for $B$, and $\$ 15$ for $C$. The production requirements and departmental capacities are as follows:
  
-What is the constraint for the assembling department?

Accepted Answer

A)  $2 \mathrm{~A}+3 \mathrm{~B}+2 \mathrm{C} \leq 30,000$ 
B)  $2 A+3 B+2 C \leq 28,000$ 
C)  $1 \mathrm{~A}+2 \mathrm{~B}+3 \mathrm{C} \leq 30,000$ 
D)  $2 A+3 B+2 B \leq 38,000$ 
A)  $2 \mathrm{~A}+3 \mathrm{~B}+2 \mathrm{C} \leq 30,000$ 
B)  $2 A+3 B+2 C \leq 28,000$ 
C)  $1 \mathrm{~A}+2 \mathrm{~B}+3 \mathrm{C} \leq 30,000$ 
D)  $2 A+3 B+2 B \leq 38,000$

Question 15

Write the word or phrase that best completes each statement or answers thequestion.
-Explain why the solution to a linear programming problem always occurs at a corner point of the feasible region.

Accepted Answer

The coordinates of a corner po

Question 16

Provide an appropriate response.
-A linear program is defined with constraints $2 x+2 y \geq 4,7 x+9 y \geq 0, x \geq 0$, and $y \geq 0$. Is the feasibility region bounded, unbounded, or empty?

Accepted Answer

A)  Bounded 
B)  Unbounded 
C)  Empty 
A)  Bounded 
B)  Unbounded 
C)  Empty

Question 17

Write the basic solution for the simplex tableau determined by setting the nonbasic variables equal to 0.
-

Accepted Answer

A) $x_{1}=6, x_{2}=0, x_{3}=13, x_{4}=18, x_{5}=1, z=11$ 
B)  $x_{1}=6, x_{2}=0, x_{3}=13, x_{4}=18, x_{5}=0, z=11$ 
C)  $x_{1}=6, x_{2}=0, x_{3}=13, x_{4}=18, x_{5}=1, z=0$ 
D) $x_{1}=0, x_{2}=-3, x_{3}=0, x_{4}=0, x_{5}=1, z=1$ 
A) $x_{1}=6, x_{2}=0, x_{3}=13, x_{4}=18, x_{5}=1, z=11$ 
B)  $x_{1}=6, x_{2}=0, x_{3}=13, x_{4}=18, x_{5}=0, z=11$ 
C)  $x_{1}=6, x_{2}=0, x_{3}=13, x_{4}=18, x_{5}=1, z=0$ 
D) $x_{1}=0, x_{2}=-3, x_{3}=0, x_{4}=0, x_{5}=1, z=1$

Question 18

Use the simplex method to solve the linear programming problem.
-Maximize $z=5 x_{1}+3 x_{2}$
Subject to: $2 x_{1}+4 x_{2} \leq 13$
$x_{1}+2 x_{2} \leq 6$
With $\quad x_{1} \geq 0, x_{2} \geq 0$

Accepted Answer

A)  Maximum is 18 when $x_{1}=0, x_{2}=6$ 
B)  Maximum is 9 when $x_{1}=0, x_{2}=3$ 
C)  Maximum is 30 when $x_{1}=6, x_{2}=0$ 
D)  Maximum is 32.5 when $\mathrm{x}_{1}=6.5, \mathrm{x}_{2}=0$ 
A)  Maximum is 18 when $x_{1}=0, x_{2}=6$ 
B)  Maximum is 9 when $x_{1}=0, x_{2}=3$ 
C)  Maximum is 30 when $x_{1}=6, x_{2}=0$ 
D)  Maximum is 32.5 when $\mathrm{x}_{1}=6.5, \mathrm{x}_{2}=0$

Question 19

Use the two-stage method to solve.
-Find $x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0$ such that
$$\begin{aligned}
& x_{1}+3 x_{2}+2 x_{3} \leq 60 \
& x_{1}+2 x_{2}+3 x_{3} \geq 36
\end{aligned}$$
And $\mathrm{z}=4 \mathrm{x}_{1}+3 \mathrm{x}_{2}+2 \mathrm{x}_{3}$ is maximized.

Accepted Answer

A)  $x_{1}=60, x_{2}=0, x_{3}=0, z=240$ 
B)  $x_{1}=50, x_{2}=5, x_{3}=0, z=215$ 
C)  $x_{3}=30, x_{1}=0, x_{2}=0, z=60$ 
D)  $x_{2}=20, x_{1}=0, x_{3}=0, z=60$ 
A)  $x_{1}=60, x_{2}=0, x_{3}=0, z=240$ 
B)  $x_{1}=50, x_{2}=5, x_{3}=0, z=215$ 
C)  $x_{3}=30, x_{1}=0, x_{2}=0, z=60$ 
D)  $x_{2}=20, x_{1}=0, x_{3}=0, z=60$

Question 20

Find the transpose of the matrix.
-

Accepted Answer

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

Provide an appropriate response. -Describe the feasible region of $x+y \geq 9$ and $x+y \leq-6$ .

Use duality to solve the problem. -Minimize $w=5 y 1+3 y_{2}$ Subject to: $2 \mathrm{y}_{1}+3 \mathrm{y}_{2} \geq 9$ $2 \mathrm{y}_{1}+\mathrm{y}_{2} \geq 11$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0$

Use the indicated entry as the pivot and perform the pivoting once. -

Provide an appropriate response. -No unique optimum solution found from a simplex tableau corresponds to

Find the value(s) of the function on the given feasible region. -Find the maximum and minimum of $z=8 x-9 y$ .

Graph the linear inequality. - $x+2 y \geq-1$ $Graph the linear inequality. - x+2 y \geq-1$

Write the word or phrase that best completes each statement or answers thequestion. -Explain how you decide which half-plane to shade when you are graphing an inequality.

Graph the linear inequality. - $2 x+5 y \leq 10$ $Graph the linear inequality. - 2 x+5 y \leq 10$

Provide an appropriate response. -If the inequalities $x \geq 0$ and $y \geq 0$ are included in a system, the feasibility region is restricted to the axes and which quadrant?

Write the word or phrase that best completes each statement or answers thequestion. -Explain why the solution to a linear programming problem always occurs at a corner point of the feasible region.

Provide an appropriate response. -A linear program is defined with constraints $2 x+2 y \geq 4,7 x+9 y \geq 0, x \geq 0$ , and $y \geq 0$ . Is the feasibility region bounded, unbounded, or empty?

Write the basic solution for the simplex tableau determined by setting the nonbasic variables equal to 0. -

Use the simplex method to solve the linear programming problem. -Maximize $z=5 x_{1}+3 x_{2}$ Subject to: $2 x_{1}+4 x_{2} \leq 13$ $x_{1}+2 x_{2} \leq 6$ With $\quad x_{1} \geq 0, x_{2} \geq 0$

Use the two-stage method to solve. -Find $x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0$ such that +3+2\leq60 +2+3\geq36 And $\mathrm{z}=4 \mathrm{x}_{1}+3 \mathrm{x}_{2}+2 \mathrm{x}_{3}$ is maximized.

Find the transpose of the matrix. -

Algebra and Equations

Graphs, Lines, and Inequalities

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Exponential and Logarithmic Functions

Mathematics of Finance

Systems of Linear Equations and Matrices

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Counting, Probability Distributions, and Further Topics in Probability

Introduction to Statistics

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Exam 7: Linear Programming

Provide an appropriate response. -Describe the feasible region of x+y≥9x+y \geq 9x+y≥9 and x+y≤−6x+y \leq-6x+y≤−6 .

Introduce slack variables as necessary and write the initial simplex tableau for the problem. -Maximize z=4x1+2x2z=4 x_{1}+2 x_{2}z=4x1​+2x2​ Subject to: 2x1+x2≤252 x_{1}+x_{2} \leq 252x1​+x2​≤25 3+5\leq74 \geq0,\geq0

Use the indicated entry as the pivot and perform the pivoting once. -

Provide an appropriate response. -No unique optimum solution found from a simplex tableau corresponds to

Find the value(s) of the function on the given feasible region. -Find the maximum and minimum of z=8x−9yz=8 x-9 yz=8x−9y .

Graph the linear inequality. - x+2y≥−1x+2 y \geq-1x+2y≥−1

Write the word or phrase that best completes each statement or answers thequestion. -Explain how you decide which half-plane to shade when you are graphing an inequality.

Graph the linear inequality. - 2x+5y≤102 x+5 y \leq 102x+5y≤10

Provide an appropriate response. -If the inequalities x≥0x \geq 0x≥0 and y≥0y \geq 0y≥0 are included in a system, the feasibility region is restricted to the axes and which quadrant?

A manufacturing company wants to maximize profits on products A,BA, BA,B , and CCC . The profit margin is $3\$ 3$3 for A,$6A, \$ 6A,$6 for BBB , and $15\$ 15$15 for CCC . The production requirements and departmental capacities are as follows: -What is the constraint for the assembling department?

Write the word or phrase that best completes each statement or answers thequestion. -Explain why the solution to a linear programming problem always occurs at a corner point of the feasible region.

Provide an appropriate response. -A linear program is defined with constraints 2x+2y≥4,7x+9y≥0,x≥02 x+2 y \geq 4,7 x+9 y \geq 0, x \geq 02x+2y≥4,7x+9y≥0,x≥0 , and y≥0y \geq 0y≥0 . Is the feasibility region bounded, unbounded, or empty?

Write the basic solution for the simplex tableau determined by setting the nonbasic variables equal to 0. -

Use the simplex method to solve the linear programming problem. -Maximize z=5x1+3x2z=5 x_{1}+3 x_{2}z=5x1​+3x2​ Subject to: 2x1+4x2≤132 x_{1}+4 x_{2} \leq 132x1​+4x2​≤13 x1+2x2≤6x_{1}+2 x_{2} \leq 6x1​+2x2​≤6 With x1≥0,x2≥0\quad x_{1} \geq 0, x_{2} \geq 0x1​≥0,x2​≥0

Use the two-stage method to solve. -Find x1≥0,x2≥0,x3≥0x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0x1​≥0,x2​≥0,x3​≥0 such that +3+2\leq60 +2+3\geq36 And z=4x1+3x2+2x3\mathrm{z}=4 \mathrm{x}_{1}+3 \mathrm{x}_{2}+2 \mathrm{x}_{3}z=4x1​+3x2​+2x3​ is maximized.

Find the transpose of the matrix. -

Algebra and Equations

Graphs, Lines, and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Mathematics of Finance

Systems of Linear Equations and Matrices

Sets and Probability

Counting, Probability Distributions, and Further Topics in Probability

Introduction to Statistics

Differential Calculus

Applications of the Derivative

Integral Calculus

Multivariate Calculus

Filters

Provide an appropriate response. -Describe the feasible region of $x+y \geq 9$ and $x+y \leq-6$ .

Find the value(s) of the function on the given feasible region. -Find the maximum and minimum of $z=8 x-9 y$ .

Graph the linear inequality. - $x+2 y \geq-1$ $Graph the linear inequality. - x+2 y \geq-1$

Graph the linear inequality. - $2 x+5 y \leq 10$ $Graph the linear inequality. - 2 x+5 y \leq 10$

Provide an appropriate response. -If the inequalities $x \geq 0$ and $y \geq 0$ are included in a system, the feasibility region is restricted to the axes and which quadrant?

Provide an appropriate response. -A linear program is defined with constraints $2 x+2 y \geq 4,7 x+9 y \geq 0, x \geq 0$ , and $y \geq 0$ . Is the feasibility region bounded, unbounded, or empty?

Use the simplex method to solve the linear programming problem. -Maximize $z=5 x_{1}+3 x_{2}$ Subject to: $2 x_{1}+4 x_{2} \leq 13$ $x_{1}+2 x_{2} \leq 6$ With $\quad x_{1} \geq 0, x_{2} \geq 0$

Use the two-stage method to solve. -Find $x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0$ such that +3+2\leq60 +2+3\geq36 And $\mathrm{z}=4 \mathrm{x}_{1}+3 \mathrm{x}_{2}+2 \mathrm{x}_{3}$ is maximized.