Minimize $c = 5 x + y + z + 2 w$ Subject to \[\begin{aligned} 5 x - y + w & \geq 1,000 \\ z + w & \leq 2,000 \\ x + y & \leq 500 \\ x \geq 0 , y \geq 0 , z \geq 0 , w & \geq 0 \end{aligned}\]

A) $c = 600$ , $x = 0$ , $y = 200$ , $z = 200$ , $w = 200$ B) $c = 1,600$ , $x = 200$ , $y = 200$ , $z = 200$ , $w = 200$ C) $c = 1,000$ , $x = 200$ , $y = 0$ , $z = 0$ , $w = 0$ D) $c = 200$ , $x = 0$ , $y = 200$ , $z = 0$ , $w = 0$ E) no solution A) $c = 600$ , $x = 0$ , $y = 200$ , $z = 200$ , $w = 200$ B) $c = 1,600$ , $x = 200$ , $y = 200$ , $z = 200$ , $w = 200$ C) $c = 1,000$ , $x = 200$ , $y = 0$ , $z = 0$ , $w = 0$ D) $c = 200$ , $x = 0$ , $y = 200$ , $z = 0$ , $w = 0$ E) no solution

Solve the linear programming problem. Maximize $P = 2 x + 5 y$ subject to $\begin{array} { l } x + 3 y \leq 24 \\ 2 x + y \leq 18 \\ 3 x + y \leq 24 \\ x \geq 0 , y \geq 0 \end{array}$ $x =$ __________; $y =$ __________; $P =$ __________

The answer of Solve the linear programming problem. Maximize \(P =...

Solve the linear programming problem. Minimize $C = - 2 x - 5 y$ subject to \[\begin{array} { l } 3 x + 5 y \leq 10 \\ 7 x - 6 y \leq 18 \\ x \geq 0 , y \geq 0 \end{array}\]

A) $x = 1$ , $y = 1$ , $C = - 7$ B) $x = 2$ , $y = 2$ , $C = - 14$ C) $x = 0$ , $y = 2$ , $C = - 10$ D) $x = 2$ , $y = 0$ , $C = - 4$ E) $x = 1$ , $y = 2$ , $C = - 12$ A) $x = 1$ , $y = 1$ , $C = - 7$ B) $x = 2$ , $y = 2$ , $C = - 14$ C) $x = 0$ , $y = 2$ , $C = - 10$ D) $x = 2$ , $y = 0$ , $C = - 4$ E) $x = 1$ , $y = 2$ , $C = - 12$

Maximize $p = 2 x + 3 y + z + 4 w$ subject to \[\begin{aligned} x + y + z + w & \leq 60 \\ 2 x + y - z - w & \geq 12 \\ x + y + z + w & \geq 12 \\ x \geq 0 , y \geq 0 , z \geq 0 , w & \geq 0 \end{aligned}\] If there is no solution, do not fill the fields. $p =$ __________ $x =$ __________ $y =$ __________ $z =$ __________ $w ^ { \prime } =$ __________

The answer of Maximize \(p = 2 x + 3...

Minimize $c = 50 x + 11 y + 50 z$ Subject to \[\begin{array} { r } 3 x + z \geq 6 \\ 3 x + y - z \geq 6 \\ 4 x + y - z \leq 8 \\ x \geq 0 , y \geq 0 , z \geq 0 \end{array}\]

A) $c = 400$ , $x = 8$ , $y = 0$ , $z = 0$ B) $c = 716$ , $x = 8$ , $y = 6$ , $z = 5$ C) $c = 88$ , $x = 0$ , $y = 0$ , $z = 8$ D) $c = 100$ , $x = 2$ , $y = 0$ , $z = 0$ E) no solution A) $c = 400$ , $x = 8$ , $y = 0$ , $z = 0$ B) $c = 716$ , $x = 8$ , $y = 6$ , $z = 5$ C) $c = 88$ , $x = 0$ , $y = 0$ , $z = 8$ D) $c = 100$ , $x = 2$ , $y = 0$ , $z = 0$ E) no solution

Write down (without solving) the dual LP problem. Maximize $p = x + y + z + w$ subject to \[\begin{array} { l } x + y + z \leq 9 \\ y + z + w \leq 7 \\ x + z + w \leq 2 \\ x + y + w \leq 4 \\ x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0 \end{array}\]

A) minimize $c = 9 s + 7 t + 2 u + 4 v$ subject to \[\begin{array} { l } s + u + v \leq 1 \\ s + t + v \leq 1 \\ s + t + u \leq 1 \\ t + u + v \leq 1 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\] B) minimize $c = s + t + u + v$ subject to \[\begin{array} { l } s + t + u \leq 7 \\ t + u + v \leq 6 \\ s + u + v \leq 3 \\ s + t + v \leq 4 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\] C) minimize $c = s + t + u + v$ subject to \[\begin{array} { l } s + t + u \geq 9 \\ t + u + v \geq 7 \\ s + u + v \geq 2 \\ s + t + v \geq 4 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\] D) minimize $c = s + t + u + v$ subject to \[\begin{array} { l } s + t + u \geq 9 \\ t + u + v \geq 7 \\ s + u + v \geq 2 \\ s + t + v \geq 4 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\] E) minimize $c = 9 s + 7 t + 2 u + 4 v$ subject to \[\begin{array} { l } s + u + v \leq 1 \\ s + t + v \leq 1 \\ s + t + u \leq 1 \\ t + u + v \leq 1 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\] A) minimize $c = 9 s + 7 t + 2 u + 4 v$ subject to \[\begin{array} { l } s + u + v \leq 1 \\ s + t + v \leq 1 \\ s + t + u \leq 1 \\ t + u + v \leq 1 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\] B) minimize $c = s + t + u + v$ subject to \[\begin{array} { l } s + t + u \leq 7 \\ t + u + v \leq 6 \\ s + u + v \leq 3 \\ s + t + v \leq 4 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\] C) minimize $c = s + t + u + v$ subject to \[\begin{array} { l } s + t + u \geq 9 \\ t + u + v \geq 7 \\ s + u + v \geq 2 \\ s + t + v \geq 4 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\] D) minimize $c = s + t + u + v$ subject to \[\begin{array} { l } s + t + u \geq 9 \\ t + u + v \geq 7 \\ s + u + v \geq 2 \\ s + t + v \geq 4 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\] E) minimize $c = 9 s + 7 t + 2 u + 4 v$ subject to \[\begin{array} { l } s + u + v \leq 1 \\ s + t + v \leq 1 \\ s + t + u \leq 1 \\ t + u + v \leq 1 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\]

Maximize $p = 12 x + 10 y$ subject to $\begin{aligned} x + y & \leq 40 \\ x & \geq 3 \\ - x + 2 y & \geq 0 \\ x \geq 0 , y & \geq 0 \end{aligned}$ Please enter your answer as three numbers, separated by commas in the order p, x, y. If there is no solution, enter no solution.

The answer of Maximize \(p = 12 x + 10...

Minimize $c = 50 x + 50 y + 11 z$ subject to \[\begin{aligned} 2 x + z & \geq 9 \\ 2 x + y - z & \geq 4 \\ 3 x + y - z & \leq 6 \\ x \geq 0 , y \geq 0 , z & \geq 0 \end{aligned}\] If there is no solution, do not fill the fields. $c =$ __________ $x =$ __________ $y =$ __________ $z =$ __________

The answer of Minimize \(c = 50 x + 50...

Minimize $c = - x + 2 y$ subject to \[\begin{aligned} y & \leq \frac { 2 x } { 5 } \\ x & \leq 5 y \\ y & \geq 4 \\ x & \geq 10 \\ x + y & \leq 24 \end{aligned}\] Min: $c =$ __________, $x =$ __________, $y =$ __________

The answer of Minimize \(c = - x + 2...

Exam 6: Linear Programming

Write down ( without solving ) the dual LP problem. Minimize $p = 5 a + 4 b$ subject to $\left\{ \begin{array} { c } a + b \geq 8 \\5 a + b \geq 20 \\a \geq 0 , b \geq 0\end{array} \right\}$ What is the dual problem in matrix form

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Question 101

Federal Rent-a-Car is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for $500,000. Admiral Motors is offering 5 small, 10 medium, and 5 large cars for $400,000. Chrysalis is offering 10 small, 5 medium, and 5 large cars for $300,000. Federal would like to buy at least 750 small cars, at least 600 medium cars, and at least 750 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible package(s) of Fred Motors, package(s) of Admiral Motors, and package(s) of Chrysalis. What will be the total cost $

(Short Answer)

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Question 102

Minimize $c = 5 x + y + z + 2 w$ Subject to 5x-y+w \geq1,000 z+w \leq2,000 x+y \leq500 x\geq0,y\geq0,z\geq0,w \geq0

(Multiple Choice)

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Question 103

Solve the standard minimization problem using duality. Minimize $c = 3 s + 3 t + 3 u$ subject to 3s+2t+u\geq60 2s+t+3u\geq60 s+3t+2u\geq60 s\geq0,t\geq0,u\geq0 $c = \quad s = \quad t = \quad u =$

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Question 104

The Miami Beach City Council has offered to subsidize hotel development in Miami Beach, and it is hoping for at least two hotels with a total capacity of at least 1,800. Suppose that you are a developer interested in taking advantage of this offer by building a small group of hotels in Miami Beach. You are thinking of three prototypes: a convention-style hotel with 500 rooms costing $100 million, a vacation-style hotel with 200 rooms costing $20 million, and a small motel with 50 rooms costing $4 million. The City Council will approve your plans, provided you build at least one convention-style hotel and no more than two small motels. How many of each type of hotel should you build to satisfy the city council's wishes and stipulations while minimizing your total cost convention-style hotel(s), vacation-style hotel(s), and __ small motel(s). Now assume that the city council will give developers 20 percent of the cost of building new hotels in Miami Beach, up to $70 million. Will the city's $70 million subsidy be sufficient to cover 20 percent of your total costs

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Question 105

In February, 2002, each episode of "Becker" was typically seen in 8.3 million homes, while each episode of "The Simpsons" was seen in 7.5 million homes. Your marketing services firm has been hired to promote Bald No More, Inc.'s, hair replacement process by buying at least 30 commercial spots during episodes of "Becker" and "The Simpsons." The cable company running "Becker" has quoted a price of $2,000 per spot, while the cable company showing "The Simpsons" has quoted a price of $1,500 per spot. Bald No More's advertising budget for TV commercials is $70,000, and it would like no more than 50% of the total number of spots to appear on "The Simpsons." How many spots should you purchase on each show to reach the most homes ?

(Multiple Choice)

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Question 106

We return to your exploits coordinating distribution for the Tubular Ride Boogie Board Company. You will recall that the company has manufacturing plants in Tucson, Arizona and Toronto, Ontario, and you have been given the job of coordinating distribution of their latest model, the Gladiator, to their outlets in Honolulu and Venice Beach. The Tucson plant can manufacture up to 620 boards per week, while the Toronto plant, beset by labor disputes, can produce no more than 410 Gladiator boards per week. The outlet in Honolulu orders 470 Gladiator boards per week, while Venice Beach orders 500 boards per week. Transportation costs are as follows: Tucson to Honolulu: $10 per board; Tucson to Venice Beach: $5 per board; Toronto to Honolulu: $20 per board; Toronto to Venice Beach: $10 per board. Your manager has said that you are to be sure to fill all orders and ship the boogie boards at a minimum total transportation cost. But you have just been notified that workers at the Toronto boogie board plant have gone on strike, resulting in a total work stoppage. You are to come up with a revised delivery schedule by tomorrow with the understanding that the Tucson plant can push production to a maximum of 810 boards per week. What should you do

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Question 107

Ruff makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat/ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 200 grams of protein and at least 50 grams of fat. If chicken costs 15 cents per ounce and grain costs 2 cents per ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost What are the shadow costs of protein and fat ?

(Multiple Choice)

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Question 108

Solve the linear programming problem. Maximize $P = 2 x + 5 y$ subject to x+3y\leq24 2x+y\leq18 3x+y\leq24 x\geq0,y\geq0 $x =$ ; $y =$ ; $P =$ __

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Question 109

Solve the linear programming problem. Maximize $P = 9 x + 6 y$ subject to $x + 3 y \leq 24$ $2 x + y \leq 18$ $3 x + y \leq 24$ $x \geq 0 , y \geq 0$

(Multiple Choice)

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Question 110

Solve the linear programming problem. Minimize $C = - 2 x - 5 y$ subject to 3x+5y\leq10 7x-6y\leq18 x\geq0,y\geq0

(Multiple Choice)

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Question 111

A region is defined by the set of inequalities. Which graph satisfies the solution to these inequalities $\left\{ \begin{array} { l } 2 x + y \leq 60 \\x + y \leq 50 \\x \geq 0 , y \geq 0\end{array} \right.$

(Multiple Choice)

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Question 112

Maximize $p = 2 x + 3 y + z + 4 w$ subject to x+y+z+w \leq60 2x+y-z-w \geq12 x+y+z+w \geq12 x\geq0,y\geq0,z\geq0,w \geq0 If there is no solution, do not fill the fields. $p =$ $x =$ $y =$ $z =$ $w ^ { \prime } =$ __

(Short Answer)

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Question 113

We return to your exploits coordinating distribution for the Tubular Ride Boogie Board Company. You will recall that the company has manufacturing plants in Tucson, Arizona and Toronto, Ontario, and you have been given the job of coordinating distribution of their latest model, the Gladiator, to their outlets in Honolulu and Venice Beach. The Tucson plant can manufacture up to 620 boards per week, while the Toronto plant, beset by labor disputes, can produce no more than 410 Gladiator boards per week. The outlet in Honolulu orders 480 Gladiator boards per week, while Venice Beach orders 510 boards per week. Transportation costs are as follows: Tucson to Honolulu: $10 per board; Tucson to Venice Beach: $5 per board; Toronto to Honolulu: $20 per board; Toronto to Venice Beach: $10 per board. Your manager has said that you are to be sure to fill all orders and ship the boogie boards at a minimum total transportation cost. But you have just been notified that workers at the Toronto boogie board plant have gone on strike, resulting in a total work stoppage. You are to come up with a revised delivery schedule by tomorrow with the understanding that the Tucson plant can push production to a maximum of 880 boards per week. Can you devise a delivery schedule with these accommodations in mind ?

(Multiple Choice)

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Question 114

Minimize $c = 50 x + 11 y + 50 z$ Subject to 3x+z\geq6 3x+y-z\geq6 4x+y-z\leq8 x\geq0,y\geq0,z\geq0

(Multiple Choice)

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Question 115

Write down (without solving) the dual LP problem. Maximize $p = x + y + z + w$ subject to x+y+z\leq9 y+z+w\leq7 x+z+w\leq2 x+y+w\leq4 x\geq0,y\geq0,z\geq0,w\geq0

(Multiple Choice)

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Question 116

Maximize $p = 12 x + 10 y$ subject to x+y \leq40 x \geq3 -x+2y \geq0 x\geq0,y \geq0 Please enter your answer as three numbers, separated by commas in the order p, x, y. If there is no solution, enter no solution.

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Question 117

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Minimize $c = 2 x - 2 y$ subject to \leqy y \leq x+y \geq7 x+2y \leq20 x\geq0,y \geq0 Min: $c =$ , $x =$ , $y =$ __

(Short Answer)

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Question 118

Minimize $c = 50 x + 50 y + 11 z$ subject to 2x+z \geq9 2x+y-z \geq4 3x+y-z \leq6 x\geq0,y\geq0,z \geq0 If there is no solution, do not fill the fields. $c =$ $x =$ $y =$ $z =$

(Short Answer)

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Question 119

Minimize $c = - x + 2 y$ subject to y \leq x \leq5y y \geq4 x \geq10 x+y \leq24 Min: $c =$ , $x =$ , $y =$ __

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Question 120

Write down ( without solving ) the dual LP problem. Minimize $p = 5 a + 4 b$ subject to $\left\{ \begin{array} { c } a + b \geq 8 \\5 a + b \geq 20 \\a \geq 0 , b \geq 0\end{array} \right\}$ What is the dual problem in matrix form

Minimize $c = 5 x + y + z + 2 w$ Subject to 5x-y+w \geq1,000 z+w \leq2,000 x+y \leq500 x\geq0,y\geq0,z\geq0,w \geq0

Solve the standard minimization problem using duality. Minimize $c = 3 s + 3 t + 3 u$ subject to 3s+2t+u\geq60 2s+t+3u\geq60 s+3t+2u\geq60 s\geq0,t\geq0,u\geq0 $c = \quad s = \quad t = \quad u =$

Solve the linear programming problem. Maximize $P = 2 x + 5 y$ subject to x+3y\leq24 2x+y\leq18 3x+y\leq24 x\geq0,y\geq0 $x =$ ; $y =$ ; $P =$ __

Solve the linear programming problem. Maximize $P = 9 x + 6 y$ subject to $x + 3 y \leq 24$ $2 x + y \leq 18$ $3 x + y \leq 24$ $x \geq 0 , y \geq 0$

Solve the linear programming problem. Minimize $C = - 2 x - 5 y$ subject to 3x+5y\leq10 7x-6y\leq18 x\geq0,y\geq0

A region is defined by the set of inequalities. Which graph satisfies the solution to these inequalities $\left\{ \begin{array} { l } 2 x + y \leq 60 \\x + y \leq 50 \\x \geq 0 , y \geq 0\end{array} \right.$

Maximize $p = 2 x + 3 y + z + 4 w$ subject to x+y+z+w \leq60 2x+y-z-w \geq12 x+y+z+w \geq12 x\geq0,y\geq0,z\geq0,w \geq0 If there is no solution, do not fill the fields. $p =$ $x =$ $y =$ $z =$ $w ^ { \prime } =$ __

Minimize $c = 50 x + 11 y + 50 z$ Subject to 3x+z\geq6 3x+y-z\geq6 4x+y-z\leq8 x\geq0,y\geq0,z\geq0

Write down (without solving) the dual LP problem. Maximize $p = x + y + z + w$ subject to x+y+z\leq9 y+z+w\leq7 x+z+w\leq2 x+y+w\leq4 x\geq0,y\geq0,z\geq0,w\geq0

Maximize $p = 12 x + 10 y$ subject to x+y \leq40 x \geq3 -x+2y \geq0 x\geq0,y \geq0 Please enter your answer as three numbers, separated by commas in the order p, x, y. If there is no solution, enter no solution.

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Minimize $c = 2 x - 2 y$ subject to \leqy y \leq x+y \geq7 x+2y \leq20 x\geq0,y \geq0 Min: $c =$ , $x =$ , $y =$ __

Minimize $c = 50 x + 50 y + 11 z$ subject to 2x+z \geq9 2x+y-z \geq4 3x+y-z \leq6 x\geq0,y\geq0,z \geq0 If there is no solution, do not fill the fields. $c =$ $x =$ $y =$ $z =$

Minimize $c = - x + 2 y$ subject to y \leq x \leq5y y \geq4 x \geq10 x+y \leq24 Min: $c =$ , $x =$ , $y =$ __

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Exam 6: Linear Programming

Minimize c=5x+y+z+2wc = 5 x + y + z + 2 wc=5x+y+z+2w Subject to 5x-y+w \geq1,000 z+w \leq2,000 x+y \leq500 x\geq0,y\geq0,z\geq0,w \geq0

Solve the standard minimization problem using duality. ​ Minimize c=3s+3t+3uc = 3 s + 3 t + 3 uc=3s+3t+3u ​ subject to 3s+2t+u\geq60 2s+t+3u\geq60 s+3t+2u\geq60 s\geq0,t\geq0,u\geq0 ​ c=s=t=u=c = \quad s = \quad t = \quad u =c=s=t=u=

Solve the linear programming problem. Maximize P=2x+5yP = 2 x + 5 yP=2x+5y subject to x+3y\leq24 2x+y\leq18 3x+y\leq24 x\geq0,y\geq0 ​ x=x =x= __________; y=y =y= __________; P=P =P= __________

Solve the linear programming problem. Maximize P=9x+6yP = 9 x + 6 yP=9x+6y subject to x+3y≤24x + 3 y \leq 24x+3y≤24 2x+y≤182 x + y \leq 182x+y≤18 3x+y≤243 x + y \leq 243x+y≤24 x≥0,y≥0x \geq 0 , y \geq 0x≥0,y≥0

Solve the linear programming problem. Minimize C=−2x−5yC = - 2 x - 5 yC=−2x−5y subject to 3x+5y\leq10 7x-6y\leq18 x\geq0,y\geq0

A region is defined by the set of inequalities. Which graph satisfies the solution to these inequalities {2x+y≤60x+y≤50x≥0,y≥0\left\{ \begin{array} { l } 2 x + y \leq 60 \\x + y \leq 50 \\x \geq 0 , y \geq 0\end{array} \right.⎩⎨⎧​2x+y≤60x+y≤50x≥0,y≥0​

Minimize c=50x+11y+50zc = 50 x + 11 y + 50 zc=50x+11y+50z Subject to 3x+z\geq6 3x+y-z\geq6 4x+y-z\leq8 x\geq0,y\geq0,z\geq0

Write down (without solving) the dual LP problem. Maximize p=x+y+z+wp = x + y + z + wp=x+y+z+w subject to x+y+z\leq9 y+z+w\leq7 x+z+w\leq2 x+y+w\leq4 x\geq0,y\geq0,z\geq0,w\geq0

Maximize ​ p=12x+10yp = 12 x + 10 yp=12x+10y ​ subject to ​ x+y \leq40 x \geq3 -x+2y \geq0 x\geq0,y \geq0 ​ Please enter your answer as three numbers, separated by commas in the order p, x, y. If there is no solution, enter no solution.

Minimize ​ c=50x+50y+11zc = 50 x + 50 y + 11 zc=50x+50y+11z ​ subject to ​ 2x+z \geq9 2x+y-z \geq4 3x+y-z \leq6 x\geq0,y\geq0,z \geq0 ​ If there is no solution, do not fill the fields. ​ c=c =c= __________ ​ x=x =x= __________ ​ y=y =y= __________ ​ z=z =z= __________

Minimize c=−x+2yc = - x + 2 yc=−x+2y subject to ​ y \leq x \leq5y y \geq4 x \geq10 x+y \leq24 ​ Min: c=c =c= __________, x=x =x= __________, y=y =y= __________

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Sets and Counting

Probability

Random Variables and Statistics

Introduction to the Derivative

Techniques of Differentiation

Applications of the Derivative

The Integral

Further Integration Techniques and Applications of the Integral

Functions of Several Variables

Trigonometric Models

Filters

Minimize $c = 5 x + y + z + 2 w$ Subject to 5x-y+w \geq1,000 z+w \leq2,000 x+y \leq500 x\geq0,y\geq0,z\geq0,w \geq0

Solve the standard minimization problem using duality. Minimize $c = 3 s + 3 t + 3 u$ subject to 3s+2t+u\geq60 2s+t+3u\geq60 s+3t+2u\geq60 s\geq0,t\geq0,u\geq0 $c = \quad s = \quad t = \quad u =$

Solve the linear programming problem. Maximize $P = 2 x + 5 y$ subject to x+3y\leq24 2x+y\leq18 3x+y\leq24 x\geq0,y\geq0 $x =$ ; $y =$ ; $P =$ __

Solve the linear programming problem. Maximize $P = 9 x + 6 y$ subject to $x + 3 y \leq 24$ $2 x + y \leq 18$ $3 x + y \leq 24$ $x \geq 0 , y \geq 0$

Solve the linear programming problem. Minimize $C = - 2 x - 5 y$ subject to 3x+5y\leq10 7x-6y\leq18 x\geq0,y\geq0

A region is defined by the set of inequalities. Which graph satisfies the solution to these inequalities $\left\{ \begin{array} { l } 2 x + y \leq 60 \\x + y \leq 50 \\x \geq 0 , y \geq 0\end{array} \right.$

Minimize $c = 50 x + 11 y + 50 z$ Subject to 3x+z\geq6 3x+y-z\geq6 4x+y-z\leq8 x\geq0,y\geq0,z\geq0

Write down (without solving) the dual LP problem. Maximize $p = x + y + z + w$ subject to x+y+z\leq9 y+z+w\leq7 x+z+w\leq2 x+y+w\leq4 x\geq0,y\geq0,z\geq0,w\geq0

Maximize $p = 12 x + 10 y$ subject to x+y \leq40 x \geq3 -x+2y \geq0 x\geq0,y \geq0 Please enter your answer as three numbers, separated by commas in the order p, x, y. If there is no solution, enter no solution.

Minimize $c = 50 x + 50 y + 11 z$ subject to 2x+z \geq9 2x+y-z \geq4 3x+y-z \leq6 x\geq0,y\geq0,z \geq0 If there is no solution, do not fill the fields. $c =$ $x =$ $y =$ $z =$

Minimize $c = - x + 2 y$ subject to y \leq x \leq5y y \geq4 x \geq10 x+y \leq24 Min: $c =$ , $x =$ , $y =$ __