Deck 10: Nonlinear Programming

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سؤال
Maximum profit is achieved everywhere the first derivative of the profit function equals zero.
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سؤال
If a nonlinear programming model consists of a single nonlinear objective function and a single linear constraint, it is called a(n) ________ optimization problem.
سؤال
The slope of a curve at its highest point equals 1.
سؤال
The first derivative of a cost function equals zero at the point V = 100. This is definitely the worst output volume for the producer to choose.
سؤال
Both linear and nonlinear programming models have the general form of an objective function subject to more than 1 constraint.
سؤال
Decision variables cannot be multiplied by each other in the objective function of a nonlinear program.
سؤال
Both linear and nonlinear programming models are examples of constrained optimization models.
سؤال
If a nonlinear program has been correctly formulated, procedures guarantee a solution.
سؤال
Constraints for nonlinear programs are usually nonlinear.
سؤال
In solving the facility location problem, the objective is to locate a centralized facility that serves customers or other facilities such that the distance traveled between the facility and customers or other facilities is minimized.
سؤال
Classical optimization is the use of calculus to determine the optimal value of a variable.
سؤال
A firm has a cost function of 3x2- 25x + 374. Without having two examples of their output volume and total cost, we cannot determine their fixed cost.
سؤال
Classical optimization uses calculus to determine the optimal values of a variable.
سؤال
The Lagrange multiplier is analogous to the dual variables in a linear programming problem.
سؤال
The slope of a curve at any point is equal to the derivative of the curves function.
سؤال
A profit function of Z = 3x2 - 12x + 5 reaches maximum profit at x = 2 units of output.
سؤال
In portfolio selection problems, risk is measured by the variance of the return on the portfolio.
سؤال
The Lagrange multiplier at the optimum gives only the instantaneous rate of change in the objective value.
سؤال
An optimal solution to a nonlinear programming problem will always occur at the boundary of the feasible solution space formed by the constraint.
سؤال
In an unconstrained nonlinear programming problem, we have a single nonlinear objective function and no constraints..
سؤال
The objective of a facility location problem is to minimize ________.
سؤال
If a nonlinear programming model consists of a single nonlinear objective function and no constraints, it is called a(n) ________ optimization problem.
سؤال
The first derivative of the fixed cost line is ________.
سؤال
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What is the derivative of the profit function for the XYZ company? Simplify the terms as much as possible.
سؤال
Assume price and demand are related by the following function: v = 100 - 2.5p. If fixed cost = $5000 and variable cost = $10, then the expression for profit is ________.
سؤال
________, a measure of correlation between returns on investment i and returns on investment j is used to reflect risk.
سؤال
Assume price and demand are related by the following function: v = 200 - p. If fixed cost = $10,000 and variable cost = $8, then the expression for profit is ________.
سؤال
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.

-What is the nonlinear profit function for the XYZ company? Simplify the terms as much as possible.
سؤال
Assume a nonlinear programming problem with a single constraint has been solved. The value of the Lagrange multiplier is $0.75 and the value of the optimal profit (Z) is $25. If the right-hand side of the constraint is increased from 38 to 42, the new value of Z will be ________.
سؤال
The distance formula of d = The distance formula of d =   will find the ________ distance between two locations.<div style=padding-top: 35px>
will find the ________ distance between two locations.
سؤال
The ________ measure of distance between two points on a set of X and Y coordinates is the hypotenuse of a right triangle.
سؤال
If a firm's profit is Z = 20p -2p2 + 40, then the optimal value of I yields a maximum profit of ________.
سؤال
The ________ the variability in an investment portfolio, the ________ the risk of the investment portfolio.
سؤال
If price and demand are related by the function v = 15 + 15p and the fixed cost is $150 while the variable cost is $5, then the expression for profit is ________.
سؤال
The ________ reflects the approximate change in the objective function resulting from a unit change in the quantity (right-hand-side) value of the constraint.
سؤال
If price and demand are related by the function v = 15 + 15p and the fixed cost is $150 while the variable cost is $5, then the profit at a price of 20 Rupees is ________.
سؤال
If a nonlinear programming problem results in profit (Z) of $50, and the Lagrange multiplier for a constraint is -2, the new profit will be ________ if the right-hand side of the constraint is increased by 1 unit.
سؤال
The ________ of the value of investment is a measure of risk.
سؤال
The dual value of a resource in a nonlinear programming model is given by the ________.
سؤال
If a firm's profit is Z = 100p -8p2 +16, then the maximum profit occurs where p = ________.
سؤال
A custom molder produces 6-ounce juice glasses and 10-ounce cocktail glasses. The per unit contribution for the juice glasses (x1) is equal to 60 - 5x1, and the per unit contribution for the cocktail glasses (x2) is 80 - 4x2. An expression for the total contribution is:

A) 20 - 4x2 - 5 x1
B) 60 x1 - 5x12 + 80x2 - 4x22
C) 80x1 - 5x122 - 4x22
D) 20 - (4x2)(5x1)
سؤال
Lush Lawns, Inc. provides a lawn fertilizer and weed control service. They are adding a special aeration treatment as a low-cost extra service option, which it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A budget of $2000 is to be used on this promotional campaign over the next quarter. Based on past experience in promoting its other services, Lush Lawns has been able to obtain an estimate of the relationship between sales and the amount spent on promotion in these two media:
s = 2x12 - 10x22 - 2x1x2 + 18x1 + 34x2
s.t. x1 + x2 = 2
Solve.
سؤال
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Sara has found an unlimited source of catnip so that is no longer a constraint. However, customer demand dictates that she produce 2.5 times more catnip balls than mice. Write the new constraint.
سؤال
Zoey's Catnip Toys faces the following relationship between price and demand: v = 2000 - 200p. The fixed cost is $500 and variable cost is $1. What price should Zoey charge to maximize profit?
سؤال
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Sara has found an unlimited source of catnip so that is no longer a constraint. However, customer demand dictates that she produce 2.5 times more catnip balls than mice. How will this impact the prices that she should charge to maximize profit?
سؤال
The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X12 + 2X22 - 8X1 - 12X2 + 34
Subject to: X1 + 2X2 = 4 lbs

-Determine the quantity of soap and shampoo that should be produced to maximize profit.
سؤال
If a firm's profit is Z = 12x - 6x2 + 30, and their minimum production level of x is equal to 0.5, then the level of x that maximizes profit is:

A) .5
B) 1
C) 1.5
D) 2
سؤال
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Write the formulation for this problem
سؤال
The slope of a curve at its highest point equals:

A) 0
B) 1
C) 2
D) 3
سؤال
A store has determined that the weekly sales of a product is related to the number of customers who visit the store and the square feet of shelf space, x, according to the following equation: -20x2 - 10C2 + 40Cx + 120x - 200. C represents the hundreds of customers who visit their store. If a store averages 200 customers per week, how many square feet of shelf space is required to maximize sales?

A) 3
B) 4
C) 5
D) 1 or 9
سؤال
Zoey's Catnip Toys faces the following relationship between price and demand: v = 2000 - 200p. The fixed cost is $500 and variable cost is $1. Write an expression for the total profit.
سؤال
Consider the curve 7x2 - 14x + 28. What is the highest point on this curve?
سؤال
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Determine the prices that Sara should charge to maximize profit.
سؤال
A store has determined that the weekly sales of a product is related to the number of customers who visit the store and the square feet of shelf space, x, according to the following equation: -20x2 - 10C2 + 40Cx + 120x - 200C + 600. C represents the hundreds of customers who visit their store. If a store averages 200 customers per week, how many square feet of shelf space is required to maximize sales?
سؤال
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What price for the ball bearings will maximize the profit?
سؤال
The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X12 + 2X22 - 8X1 - 12X2 + 34
Subject to: X1 + 2X2 = 4 lbs

-Determine the profit for the optimal production quantities of soap and shampoo.
سؤال
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What is the optimal production quantity?
سؤال
Consider the curve 7x2 - 14x + 28. What is the slope at x = 5?
سؤال
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What is the optimal profit?
سؤال
Consider the curve 7x2 - 14x + 28. What is the second derivative at x = 10?
سؤال
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Write the appropriate expression for the demand constraint.

A) 2.5x1 = x2
B) x1 - 2.5x2 ? 0
C) x1 + 2.5x2 ? 0
D) x1 = 2.5x2
سؤال
The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X12 + 2X22 - 8X1 - 12X2 + 34
Subject to: X1 + 2X2 = 4 lbs

-What is profit when the optimal values of soap and shampoo are produced?

A) Z = 12
B) Z = 14
C) Z = 16
D) Z = 18
سؤال
The Lagrange multiplier is

A) the shadow price for the constraint coefficients.
B) valid over a range of changes in the RHS.
C) the rate of change in the objective value as the RHS of the constraint increases.
D) the minimum threshold for decision variables to enter the solution.
سؤال
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation:
V = 1000 - 8p. What is the optimal price of the ball bearings that will maximize the profit?

A) $47.99
B) $53.99
C) $58.99
D) $63.99
سؤال
Both linear and nonlinear programming models are examples of

A) goal programming models.
B) simplex tableaus.
C) constrained likelihood models.
D) constrained optimization models.
سؤال
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Write the appropriate expression for profit.

A) Max Z = (p1 - 3)x1 + (p1 - 2)x2
B) Max Z = (p1 + 1)x1 + (p1 + 3)x2
C) Max Z = (p1 - 1)x1 + (p1 - 3)x2
D) Min Z = (p1 -1)x1 + (p1 - 3)x2
سؤال
The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X12 + 2X22 - 8X1 - 12X2 + 34
Subject to: X1 + 2X2 = 4 lbs

-What quantities of soap and shampoo maximize profit?

A) X1 = 0, X2 = 2
B) X1 = 1, X2 = 2
C) X1 = 2, X2 = 2
D) X1 = 2, X2 = 1
سؤال
The Lagrange multiplier reflects the appropriate change in the objective function resulting from a unit change in the ________ of the constraint equation.

A) coefficient
B) objective function
C) right-hand side
D) shadow price
سؤال
The Lagrange multiplier is ________ to the dual variables in a linear programming problem.

A) analogous
B) contradictory
C) inversely related
D) opposite
سؤال
The derivative of a function ________ the slope of the curve defined by that function.

A) is larger than
B) equals
C) is smaller than
D) is similar to
سؤال
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation:
V = 1000 - 8p. What is optimal profit?

A) $6668
B) $7668
C) $8668
D) $9768
سؤال
Classical optimization is the use of ________ to determine the optimal value of a variable.

A) calculus
B) linear programming
C) nonlinear programming
D) goal programming
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Deck 10: Nonlinear Programming
1
Maximum profit is achieved everywhere the first derivative of the profit function equals zero.
False
2
If a nonlinear programming model consists of a single nonlinear objective function and a single linear constraint, it is called a(n) ________ optimization problem.
constrained or nonlinear
3
The slope of a curve at its highest point equals 1.
False
4
The first derivative of a cost function equals zero at the point V = 100. This is definitely the worst output volume for the producer to choose.
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5
Both linear and nonlinear programming models have the general form of an objective function subject to more than 1 constraint.
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6
Decision variables cannot be multiplied by each other in the objective function of a nonlinear program.
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7
Both linear and nonlinear programming models are examples of constrained optimization models.
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8
If a nonlinear program has been correctly formulated, procedures guarantee a solution.
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9
Constraints for nonlinear programs are usually nonlinear.
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10
In solving the facility location problem, the objective is to locate a centralized facility that serves customers or other facilities such that the distance traveled between the facility and customers or other facilities is minimized.
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11
Classical optimization is the use of calculus to determine the optimal value of a variable.
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12
A firm has a cost function of 3x2- 25x + 374. Without having two examples of their output volume and total cost, we cannot determine their fixed cost.
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13
Classical optimization uses calculus to determine the optimal values of a variable.
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14
The Lagrange multiplier is analogous to the dual variables in a linear programming problem.
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15
The slope of a curve at any point is equal to the derivative of the curves function.
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16
A profit function of Z = 3x2 - 12x + 5 reaches maximum profit at x = 2 units of output.
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17
In portfolio selection problems, risk is measured by the variance of the return on the portfolio.
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18
The Lagrange multiplier at the optimum gives only the instantaneous rate of change in the objective value.
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19
An optimal solution to a nonlinear programming problem will always occur at the boundary of the feasible solution space formed by the constraint.
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20
In an unconstrained nonlinear programming problem, we have a single nonlinear objective function and no constraints..
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21
The objective of a facility location problem is to minimize ________.
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22
If a nonlinear programming model consists of a single nonlinear objective function and no constraints, it is called a(n) ________ optimization problem.
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23
The first derivative of the fixed cost line is ________.
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24
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What is the derivative of the profit function for the XYZ company? Simplify the terms as much as possible.
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25
Assume price and demand are related by the following function: v = 100 - 2.5p. If fixed cost = $5000 and variable cost = $10, then the expression for profit is ________.
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26
________, a measure of correlation between returns on investment i and returns on investment j is used to reflect risk.
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27
Assume price and demand are related by the following function: v = 200 - p. If fixed cost = $10,000 and variable cost = $8, then the expression for profit is ________.
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28
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.

-What is the nonlinear profit function for the XYZ company? Simplify the terms as much as possible.
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29
Assume a nonlinear programming problem with a single constraint has been solved. The value of the Lagrange multiplier is $0.75 and the value of the optimal profit (Z) is $25. If the right-hand side of the constraint is increased from 38 to 42, the new value of Z will be ________.
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30
The distance formula of d = The distance formula of d =   will find the ________ distance between two locations.
will find the ________ distance between two locations.
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31
The ________ measure of distance between two points on a set of X and Y coordinates is the hypotenuse of a right triangle.
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32
If a firm's profit is Z = 20p -2p2 + 40, then the optimal value of I yields a maximum profit of ________.
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33
The ________ the variability in an investment portfolio, the ________ the risk of the investment portfolio.
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34
If price and demand are related by the function v = 15 + 15p and the fixed cost is $150 while the variable cost is $5, then the expression for profit is ________.
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35
The ________ reflects the approximate change in the objective function resulting from a unit change in the quantity (right-hand-side) value of the constraint.
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36
If price and demand are related by the function v = 15 + 15p and the fixed cost is $150 while the variable cost is $5, then the profit at a price of 20 Rupees is ________.
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37
If a nonlinear programming problem results in profit (Z) of $50, and the Lagrange multiplier for a constraint is -2, the new profit will be ________ if the right-hand side of the constraint is increased by 1 unit.
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38
The ________ of the value of investment is a measure of risk.
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39
The dual value of a resource in a nonlinear programming model is given by the ________.
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40
If a firm's profit is Z = 100p -8p2 +16, then the maximum profit occurs where p = ________.
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41
A custom molder produces 6-ounce juice glasses and 10-ounce cocktail glasses. The per unit contribution for the juice glasses (x1) is equal to 60 - 5x1, and the per unit contribution for the cocktail glasses (x2) is 80 - 4x2. An expression for the total contribution is:

A) 20 - 4x2 - 5 x1
B) 60 x1 - 5x12 + 80x2 - 4x22
C) 80x1 - 5x122 - 4x22
D) 20 - (4x2)(5x1)
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42
Lush Lawns, Inc. provides a lawn fertilizer and weed control service. They are adding a special aeration treatment as a low-cost extra service option, which it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A budget of $2000 is to be used on this promotional campaign over the next quarter. Based on past experience in promoting its other services, Lush Lawns has been able to obtain an estimate of the relationship between sales and the amount spent on promotion in these two media:
s = 2x12 - 10x22 - 2x1x2 + 18x1 + 34x2
s.t. x1 + x2 = 2
Solve.
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43
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Sara has found an unlimited source of catnip so that is no longer a constraint. However, customer demand dictates that she produce 2.5 times more catnip balls than mice. Write the new constraint.
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44
Zoey's Catnip Toys faces the following relationship between price and demand: v = 2000 - 200p. The fixed cost is $500 and variable cost is $1. What price should Zoey charge to maximize profit?
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45
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Sara has found an unlimited source of catnip so that is no longer a constraint. However, customer demand dictates that she produce 2.5 times more catnip balls than mice. How will this impact the prices that she should charge to maximize profit?
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46
The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X12 + 2X22 - 8X1 - 12X2 + 34
Subject to: X1 + 2X2 = 4 lbs

-Determine the quantity of soap and shampoo that should be produced to maximize profit.
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47
If a firm's profit is Z = 12x - 6x2 + 30, and their minimum production level of x is equal to 0.5, then the level of x that maximizes profit is:

A) .5
B) 1
C) 1.5
D) 2
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48
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Write the formulation for this problem
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49
The slope of a curve at its highest point equals:

A) 0
B) 1
C) 2
D) 3
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50
A store has determined that the weekly sales of a product is related to the number of customers who visit the store and the square feet of shelf space, x, according to the following equation: -20x2 - 10C2 + 40Cx + 120x - 200. C represents the hundreds of customers who visit their store. If a store averages 200 customers per week, how many square feet of shelf space is required to maximize sales?

A) 3
B) 4
C) 5
D) 1 or 9
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51
Zoey's Catnip Toys faces the following relationship between price and demand: v = 2000 - 200p. The fixed cost is $500 and variable cost is $1. Write an expression for the total profit.
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52
Consider the curve 7x2 - 14x + 28. What is the highest point on this curve?
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53
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Determine the prices that Sara should charge to maximize profit.
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54
A store has determined that the weekly sales of a product is related to the number of customers who visit the store and the square feet of shelf space, x, according to the following equation: -20x2 - 10C2 + 40Cx + 120x - 200C + 600. C represents the hundreds of customers who visit their store. If a store averages 200 customers per week, how many square feet of shelf space is required to maximize sales?
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55
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What price for the ball bearings will maximize the profit?
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56
The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X12 + 2X22 - 8X1 - 12X2 + 34
Subject to: X1 + 2X2 = 4 lbs

-Determine the profit for the optimal production quantities of soap and shampoo.
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57
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What is the optimal production quantity?
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58
Consider the curve 7x2 - 14x + 28. What is the slope at x = 5?
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59
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What is the optimal profit?
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60
Consider the curve 7x2 - 14x + 28. What is the second derivative at x = 10?
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61
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Write the appropriate expression for the demand constraint.

A) 2.5x1 = x2
B) x1 - 2.5x2 ? 0
C) x1 + 2.5x2 ? 0
D) x1 = 2.5x2
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62
The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X12 + 2X22 - 8X1 - 12X2 + 34
Subject to: X1 + 2X2 = 4 lbs

-What is profit when the optimal values of soap and shampoo are produced?

A) Z = 12
B) Z = 14
C) Z = 16
D) Z = 18
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63
The Lagrange multiplier is

A) the shadow price for the constraint coefficients.
B) valid over a range of changes in the RHS.
C) the rate of change in the objective value as the RHS of the constraint increases.
D) the minimum threshold for decision variables to enter the solution.
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64
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation:
V = 1000 - 8p. What is the optimal price of the ball bearings that will maximize the profit?

A) $47.99
B) $53.99
C) $58.99
D) $63.99
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65
Both linear and nonlinear programming models are examples of

A) goal programming models.
B) simplex tableaus.
C) constrained likelihood models.
D) constrained optimization models.
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66
Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x1 = 1800 - 150p1
x2 = 1500 - 300p2
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Write the appropriate expression for profit.

A) Max Z = (p1 - 3)x1 + (p1 - 2)x2
B) Max Z = (p1 + 1)x1 + (p1 + 3)x2
C) Max Z = (p1 - 1)x1 + (p1 - 3)x2
D) Min Z = (p1 -1)x1 + (p1 - 3)x2
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67
The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X12 + 2X22 - 8X1 - 12X2 + 34
Subject to: X1 + 2X2 = 4 lbs

-What quantities of soap and shampoo maximize profit?

A) X1 = 0, X2 = 2
B) X1 = 1, X2 = 2
C) X1 = 2, X2 = 2
D) X1 = 2, X2 = 1
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68
The Lagrange multiplier reflects the appropriate change in the objective function resulting from a unit change in the ________ of the constraint equation.

A) coefficient
B) objective function
C) right-hand side
D) shadow price
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69
The Lagrange multiplier is ________ to the dual variables in a linear programming problem.

A) analogous
B) contradictory
C) inversely related
D) opposite
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70
The derivative of a function ________ the slope of the curve defined by that function.

A) is larger than
B) equals
C) is smaller than
D) is similar to
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71
The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation:
V = 1000 - 8p. What is optimal profit?

A) $6668
B) $7668
C) $8668
D) $9768
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72
Classical optimization is the use of ________ to determine the optimal value of a variable.

A) calculus
B) linear programming
C) nonlinear programming
D) goal programming
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