Question 1

An iso-profit line
A)can be used to help solve a profit maximizing linear programming problem.
B)is parallel to all other iso-profit lines in the same problem.
C)is a line with the same profit at all points.
D)all of the above.
E)none of the above.

Accepted Answer

An iso-profit line is a line that connects all the possible combinations of inputs that yield the same amount of profit. It can be used to help solve a profit maximizing linear programming problem by identifying the optimal combination of inputs that will maximize profit. Iso-profit lines are parallel to each other because they have the same profit. Therefore, they are equally spaced from each other. So, the answer is D- All of the above.

Question 2

If cars (C sell for $500 profit and trucks (T)sell for $300 profit which of the following represents the objective function?
A)Maximize = 500C + 300T
B)Minimize = 500C + 300T
C)Maximize = 500C - 300T
D)Minimize = 300T- 500C
E)Minimize = 500T - 300C

Accepted Answer

The objective is to maximize profit, so we want to maximize the total profit from selling cars and trucks. The profit from cars is $500 and the profit from trucks is $300, so the objective function is Maximize = 500C + 300T.

Question 3

A linear programming problem has two constraints 2X + 4Y = 100 and 1X + 8Y ≤ 100,plus non-negativity constraints on X and Y.Which of the following statements about its feasible region is true?
A)The points (100,0)and (0,25)both lie outside the feasible region.
B)The two corner points are (33-1/3,8-1/3)and (50,0).
C)The graphical origin (0,0)is not in the feasible region.
D)The feasible region is a straight line segment,not an area.
E)All of the above are true.

Accepted Answer

The answer of A linear programming problem has two constraints...

Question 4

Which of the following is not a requirement of a linear programming problem?
A)an objective function,expressed in terms of linear equations
B)constraint equations,expressed as linear equations
C)an objective function,to be maximized or minimized
D)alternative courses of action
E)for each decision variable,there must be one constraint or resource limit

Accepted Answer

Linear programming problems require an objective function (which can be either maximized or minimized), constraint equations (which must be linear), and alternative courses of action. However, there is no requirement that each decision variable must have its own unique constraint or resource limit. Constraints typically apply to combinations of variables, not necessarily on a one-to-one basis with variables.

Question 5

What combination of x and y will yield the optimum for this problem? Minimize $3x + $15y,subject to (1)2x + 4y < 12 and (2)5x + 2y < 10 and (3)x,y ≥ 0.
A)x = 2,y = 0
B)x = 0,y = 3
C)x = 0,y = 0
D)x = 1,y = 5
E)x = 0,y = 4

Accepted Answer

The answer of What combination of x and y will...

Question 6

The feasible region in the diagram below is consistent with which one of the following constraints? 
A)8X1 + 4X2 ≥ 160
B)4X1 + 8X2 ≤ 160
C)8X1 - 4X2 ≤ 160
D)8X1 + 4X2 ≤ 160
E)4X1 - 8X2 ≤ 160

Accepted Answer

The feasible region is bounded by the lines 8X1 + 4X2 = 160, X1 = 0, and X2 = 0.
We can see that the line 8X1 + 4X2 ≤ 160 also passes through this feasible region, as it is below the line 8X1 + 4X2 = 160. Therefore, D is the correct constraint that is consistent with this feasible region.

Question 7

Which of the following sets of constraints results in an unbounded maximizing problem? A)X + Y > 100 and X + Y < 50 B)X + Y > 15 and X - Y < 10 C)X + Y < 10 and X > 5 D)X < 10 and Y < 20 E)All of the above have a bounded maximum.

Accepted Answer

The answer of Which of the following sets of constraints...

Question 8

A linear programming problem has two constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100,plus nonnegativity constraints on X and Y.Which of the following statements about its feasible region is true?
A)There are four corner points including (50,0)and (0,12.5).
B)The two corner points are (0,0)and (50,12.5).
C)The graphical origin (0,0)is not in the feasible region.
D)The feasible region includes all points that satisfy one constraint,the other,or both.
E)The feasible region cannot be determined without knowing whether the problem is to be minimized or maximized.

Accepted Answer

The feasible region of a linear programming problem is defined by the intersection of all constraints, including nonnegativity constraints. The given constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, along with nonnegativity constraints on X and Y, define a polygonal feasible region on the XY-plane. The origin (0,0) always satisfies the given constraints and nonnegativity, making it a part of the feasible region. The points (50, 0) and (0, 12.5) are found by setting Y and X to zero in the respective constraints, showing they are corner points of the feasible region. The feasible region is determined by the intersection of the constraints, not by the objective function (whether to minimize or maximize), making E incorrect. The statement that the feasible region includes all points that satisfy one constraint, the other, or both (D) is misleading because points must satisfy all constraints simultaneously to be in the feasible region. Therefore, the correct statement is that there are four corner points including (50, 0) and (0, 12.5), which is a characteristic of the feasible region defined by the given constraints and nonnegativity conditions.

Question 9

For the constraints given below,which point is in the feasible region of this maximization problem? (1)14x + 6y < 42 (2)x - y < 3 (3)x,y ≥ 0
A)x = 2,y = 1
B)x = 1,y = 5
C)x = -1,y = 1
D)x = 4,y = 4
E)x = 2,y = 8

Accepted Answer

Substituting the values of x and y in the constraints:
(1) 14(2) + 6(1) = 34 ≤ 42
(2) 2 - 1 = 1 ≤ 3
(3) 2, 1 ≥ 0
All constraints are satisfied, and thus point (2,1) is in the feasible region.

Question 10

The corner-point solution method requires
A)identifying the corner of the feasible region that has the sharpest angle.
B)moving the iso-profit line to the highest level that still touches some part of the feasible region.
C)moving the iso-profit line to the lowest level that still touches some part of the feasible region.
D)finding the coordinates at each corner of the feasible solution space.
E)none of the above.

Accepted Answer

The corner-point solution method requires finding the coordinates at each corner of the feasible solution space, which is option D.

Question 11

What combination of x and y will yield the optimum for this problem? Maximize $3x + $15y,subject to (1)2x + 4y < 12 and (2)5x + 2y < 10 and (3)x,y ≥ 0.
A)x = 2,y = 0
B)x = 0,y = 3
C)x = 0,y = 0
D)x = 1,y = 5
E)none of the above

Accepted Answer

To find the optimum, we can graph the constraints and identify the feasible region. The objective function is maximized at a vertex of the feasible region. Testing the vertices (which are the intersections of the constraints), we find that the option B (x = 0, y = 3) satisfies all constraints and maximizes the objective function $3x + $15y, yielding the maximum value of $45.

Question 12

A maximizing linear programming problem has two constraints: 2X + 4Y < 100 and 3X + 10Y < 210,in addition to constraints stating that both X and Y must be nonnegative.The corner points of the feasible region of this problem are
A)(0,0), (50,0), (0,21),and (20,15).
B)(0,0), (70,0), (25,0),and (15,20).
C)(20,15).
D)(0,0), (0,100),and (210,0).
E)none of the above.

Accepted Answer

To find the corner points, we need to solve for the intersection of the two constraint lines. 
Setting 2X + 4Y = 100 and 3X + 10Y = 210 equal to each other and solving for X and Y gives (50,0). 
Setting 2X + 4Y = 100 and Y = 5 equal to each other gives (0, 25/2) which is not in the feasible region since Y has to be nonnegative. 
Setting 3X + 10Y = 210 and X = 0 equal to each other gives (0, 21). 
Setting 3X + 10Y = 210 and 2X + 4Y = 100 equal to each other and solving for X and Y gives (20, 15).
Therefore, the corner points of the feasible region are (0,0), (50,0), (0,21),and (20,15).

Question 13

For the two constraints given below,which point is in the feasible region of this minimization problem? (1)14x + 6y > 42 (2)x - y > 3
A)x = -1,y = 1
B)x = 0,y = 4
C)x = 2,y = 1
D)x = 5,y = 1
E)x = 2,y = 0

Accepted Answer

The answer of For the two constraints given below,which point...

Question 14

The region which satisfies all of the constraints in graphical linear programming is called the
A)area of optimal solutions.
B)area of feasible solutions.
C)profit maximization space.
D)region of optimality.
E)region of non-negativity.

Accepted Answer

The region which satisfies all of the constraints in graphical linear programming is called the area of feasible solutions. It includes all possible solutions that meet the requirements of the problem. The optimal solution(s) will be located within this region. Therefore, option (B) best describes the region. Option (A) is incorrect because while the optimal solution(s) may be in the feasible region, the feasible region itself is not designated as such. Option (C) is incorrect because it only applies to a specific type of problem (profit maximization) and is not a general term used to describe the feasible region. Option (D) is incorrect because while the feasible region may contain optimal solutions, it is not designated as the region of optimality. Option (E) is incorrect because it only applies to problems involving non-negative variables and is not a general term used to describe the feasible region.

Question 15

In linear programming,a statement such as "maximize contribution" becomes a(n)
A)constraint.
B)slack variable.
C)objective function.
D)violation of linearity.
E)decision variable.

Accepted Answer

In linear programming, the objective function is the statement of what we want to maximize or minimize. In this case, "maximize contribution" is the objective function. Constraints are limitations or restrictions on the variables in the problem, slack variables are used to convert inequality constraints into equality constraints, decision variables are the variables we are trying to optimize, and violation of linearity would refer to the problem not meeting the requirements of linear programming.

Question 16

Which of the following combinations of constraints has no feasible region? A)X + Y > 15 and X - Y < 10 B)X + Y > 5 and X > 10 C)X > 10 and Y > 20 D)X + Y > 100 and X + Y < 50 E)All of the above have a feasible region.

Accepted Answer

The constraints X + Y > 100 and X + Y < 50 cannot both be true at the same time. Therefore, there is no feasible region for this set of constraints.

Question 17

A linear programming problem has two constraints 2X + 4Y ≥ 100 and 1X + 8Y ≤ 100,plus nonnegativity constraints on X and Y.Which of the following statements about its feasible region is true?
A)There are four corner points including (50,0)and (0,12.5).
B)The two corner points are (0,0)and (50,12.5).
C)The graphical origin (0,0)is in the feasible region.
D)The feasible region is triangular in shape,bounded by (50,0), (33-1/3,8-1/3),and (100,0).
E)The feasible region cannot be determined without knowing whether the problem is to be minimized or maximized.

Accepted Answer

To graph the feasible region, we first graph the two lines:
2X + 4Y = 100 
1X + 8Y = 100
The intercepts are as follows;
2X+4Y=100 => Y= 25-1/2X. Intercept of Y=25 and X=0, and Y=0 and X=50
1X+8Y=100=> Y= 12.5 - 1/8X. Intercept of Y=12.5 and X=0 and Y=0 and X= 100
We then shade the region that satisfies both constraints (above the first line and below the second line), which gives us a triangular feasible region bounded by (50,0), (33-1/3,8-1/3),and (100,0). Therefore, the correct answer is D.

Question 18

The feasible region in the diagram below is consistent with which one of the following constraints? 
A)8X1 + 4X2 ≤ 160
B)8X1 + 4X2 ≥ 160
C)4X1 + 8X2 ≤ 160
D)8X1 - 4X2 ≤ 160
E)4X1 - 8X2 ≤ 160

Accepted Answer

The answer of The feasible region in the diagram below...

Question 19

Using the iso-profit line solution method to solve a maximization problem requires that we
A)find the value of the objective function at the origin.
B)move the iso-profit line away from the origin until it barely touches some part of the feasible region.
C)move the iso-cost line to the lowest level that still touches some part of the feasible region.
D)test the objective function value of every corner point in the feasible region.
E)none of the above.

Accepted Answer

The iso-profit line is moved away from the origin until it barely touches the feasible region. This point of tangency is the optimal solution. The iso-cost line is not used in the iso-profit line solution method. The objective function value at the origin is irrelevant in this method. Testing every corner point in the feasible region is not necessary with this method.

Question 20

What combination of a and b will yield the optimum for this problem? Maximize $6a + $15b,subject to (1)4a + 2b < 12 and (2)5a + 2b < 20 and (3)a,b ≥ 0.
A)a = 0,b = 0
B)a = 3,b = 3
C)a = 0,b = 6
D)a = 6,b = 0
E)cannot solve without values for a and b

Accepted Answer

The answer of What combination of a and b will...

Quiz 22: Linear Programming

An iso-profit line

If cars (C sell for $500 profit and trucks (T) sell for $300 profit which of the following represents the objective function?

A linear programming problem has two constraints 2X + 4Y = 100 and 1X + 8Y ≤ 100,plus non-negativity constraints on X and Y.Which of the following statements about its feasible region is true?

Which of the following is not a requirement of a linear programming problem?

What combination of x and y will yield the optimum for this problem? Minimize $3x + $15y,subject to (1) 2x + 4y < 12 and (2) 5x + 2y < 10 and (3) x,y ≥ 0.

The feasible region in the diagram below is consistent with which one of the following constraints?

Which of the following sets of constraints results in an unbounded maximizing problem?

A linear programming problem has two constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100,plus nonnegativity constraints on X and Y.Which of the following statements about its feasible region is true?

For the constraints given below,which point is in the feasible region of this maximization problem? (1) 14x + 6y < 42 (2) x - y < 3 (3) x,y ≥ 0

The corner-point solution method requires

What combination of x and y will yield the optimum for this problem? Maximize $3x + $15y,subject to (1) 2x + 4y < 12 and (2) 5x + 2y < 10 and (3) x,y ≥ 0.

A maximizing linear programming problem has two constraints: 2X + 4Y < 100 and 3X + 10Y < 210,in addition to constraints stating that both X and Y must be nonnegative.The corner points of the feasible region of this problem are

For the two constraints given below,which point is in the feasible region of this minimization problem? (1) 14x + 6y > 42 (2) x - y > 3

The region which satisfies all of the constraints in graphical linear programming is called the

In linear programming,a statement such as "maximize contribution" becomes a(n)

Which of the following combinations of constraints has no feasible region?

A linear programming problem has two constraints 2X + 4Y ≥ 100 and 1X + 8Y ≤ 100,plus nonnegativity constraints on X and Y.Which of the following statements about its feasible region is true?

The feasible region in the diagram below is consistent with which one of the following constraints?

Using the iso-profit line solution method to solve a maximization problem requires that we

What combination of a and b will yield the optimum for this problem? Maximize $6a + $15b,subject to (1) 4a + 2b < 12 and (2) 5a + 2b < 20 and (3) a,b ≥ 0.