Question 1

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 2

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 3

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 4

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...

Question 5

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 of feasible solutions is the set of all points that satisfy all of the constraints in linear programming. It represents the feasible solutions that can be obtained by the objective function. The objective is to find the optimal solution, which is a point in the feasible region that maximizes or minimizes the objective function. Therefore, option B is the correct answer.

Question 6

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 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

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 mathematical expression used to determine the optimal solution. It outlines the goal of the problem and is typically defined as either maximizing or minimizing some quantity, such as profit or cost. Therefore, "maximize contribution" becomes the objective function in this case. The other choices are not appropriate as constraints, slack variables, violations of linearity, and decision variables are all different components of a linear programming problem.

Question 9

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 where the shaded region intersects with the coordinate axes. Since the feasible region is below the line and to the left of the y-axis, we know that the constraint is of the form aX1 + bX2 ≤ c. The only option that fits this form is D: 8X1 + 4X2 ≤ 160.

Question 10

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 11

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 solution method involves drawing a line that represents a constant level of profit and moving it away from the origin until it just touches the feasible region. This is done to find the highest level of profit that can be achieved while still staying within the feasible region. The objective function value at the origin and testing the objective function value at every corner point of the feasible region may be useful steps, but they are not required for the iso-profit line solution method. Similarly, iso-cost lines may be useful for solving cost minimization problems but are not necessary for maximizing profit. Therefore, the best choice is B as it accurately describes the main step of the iso-profit line solution method.

Question 12

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 function is usually a linear equation that represents the goal or objective of the problem. The goal in this case is to maximize profit, which is given by 500C + 300T where C represents the number of cars sold and T represents the number of trucks sold. Therefore, the correct choice is A, which represents the objective function as maximizing 500C + 300T.

Question 13

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 represents all combinations of inputs that result in the same profit. It is used in solving profit maximizing linear programming problems because it helps to identify the optimal combination of inputs that will result in the highest profit. It is parallel to all other iso-profit lines because they all represent the same level of profit. Therefore, the correct option is D, as all the options are correct.

Question 14

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 15

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 involves finding the coordinates at each corner of the feasible solution space, which are the points where the constraints meet. These coordinates are then evaluated to determine which corner point gives the optimal solution. Therefore, choice D is the correct answer. Choices A, B, and C are not accurate descriptions of the corner-point solution method.

Question 16

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 plot the two lines given by the inequality constraints and determine where they intersect.

2X + 4Y = 100 can be simplified to Y = -0.5X + 25, which has a Y-intercept of 25 and a slope of -0.5.

3X + 10Y = 210 can be simplified to Y = (-0.3)X + 21, which has a Y-intercept of 21 and a slope of -0.3.

Plotting these lines and determining where they intersect, we find the following corner points:

(0,0): This point is where the two lines intersect at the origin. 
(50,0): This point is where the first line intersects the X-axis. Solving the equation for 2X + 4Y = 100 for X when Y=0 gives X=50. 
(0,21): This point is where the second line intersects the Y-axis. Solving the equation for 3X + 10Y = 210 for Y when X=0 gives Y=21. 
(20,15): This point is the intersection of the two lines when they are neither parallel nor coincident, which is the feasible region of this problem. Solving the system of equations for 2X + 4Y = 100 and 3X + 10Y = 210 gives X=20 and Y=15.

Therefore, the corner points of the feasible region are (0,0), (50,0), (0,21), and (20,15). The answer is A.

Question 17

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 18

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 find the feasible region, we graph the two constraints on the same coordinate axes. The shaded area where both inequalities are satisfied is the feasible region.

2X + 4Y ≥ 100 can be simplified to X + 2Y ≥ 50, or Y ≥ -0.5X + 25.
1X + 8Y ≤ 100 can be simplified to Y ≤ -0.125X + 12.5.

Graphing these two inequalities, we get a triangular feasible region bounded by (50, 0), (33-1/3, 8-1/3), and (100, 0). Therefore, statement D is true.

Statement A is false since there are only three corner points in this feasible region. Statement B is false since (0, 0) is not a corner point. Statement C is false since the origin is not in the feasible region. Statement E is false since the shape of the feasible region is independent of whether we are minimizing or maximizing.

Question 19

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 20

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

Constraints X + Y > 100 and X + Y < 50 cannot both be true simultaneously. Therefore, there is no feasible region for this combination of constraints. The other choices have feasible regions.

Quiz 22: Linear Programming

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

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

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

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.

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

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?

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

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

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

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.

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

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

An iso-profit line

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 corner-point solution method requires

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 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 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?

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

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