Minimize $c = 2.2 x + y + 3.3 z$ subject to \[\begin{aligned} x + 1.5 y + 1.2 z & \geq 100 \\ 2 x + 1.5 y & \geq 40 \\ 1.5 y + 1.1 z & \geq 40 \\ x \geq 0 , y \geq 0 , z & \geq 0 \end{aligned}\] Round answers to two decimal places, if necessary. If there is no solution, do not fill the fields. $c =$ __________ $x =$ __________ $y =$ __________ $z =$ __________

The answer of Minimize \(c = 2.2 x + y...

Maximize $p = 12 x + 10 y$ Subject to \[\begin{aligned} x + y & \leq 40 \\ x & \geq 1 \\ - x + 2 y & \geq 0 \\ x \geq 0 , y & \geq 0 \end{aligned}\]

A) $p = 1,360$ , $x = 80$ , $y = 40$ B) $p = \frac { 560 } { 3 }$ , $x = \frac { 40 } { 3 }$ , $y = \frac { 80 } { 3 }$ C) $p = \frac { 1,360 } { 3 } , x = \frac { 80 } { 3 } , y = \frac { 40 } { 3 }$ D) $p = 1,280$ , $x = 40$ , $y = 80$ E) no solution A) $p = 1,360$ , $x = 80$ , $y = 40$ B) $p = \frac { 560 } { 3 }$ , $x = \frac { 40 } { 3 }$ , $y = \frac { 80 } { 3 }$ C) $p = \frac { 1,360 } { 3 } , x = \frac { 80 } { 3 } , y = \frac { 40 } { 3 }$ D) $p = 1,280$ , $x = 40$ , $y = 80$ E) no solution

Maximize $p = 2 x + 3 y + z + 4 w$ Subject to $\begin{aligned} x + y + z + w & \leq 40 \\ 2 x + y - z - w & \geq 10 \\ x + y + z + w & \geq 10 \\ x \geq 0 , y \geq 0 , z \geq 0 , w & \geq 0 \end{aligned}$

A) $p = 65$ , $x = 25$ , $y = 0$ , $z = 15$ , $w = 0$ B) $p = 135$ , $x = 0$ , $y = 25$ , $z = 0$ , $w = 15$ C) $p = 130$ , $x = 0$ , $y = 15$ , $z = 0$ , $w = 15$ D) $p = 55$ , $x = 15$ , $y = 0$ , $z = 25$ , $w = 0$ E) no solution A) $p = 65$ , $x = 25$ , $y = 0$ , $z = 15$ , $w = 0$ B) $p = 135$ , $x = 0$ , $y = 25$ , $z = 0$ , $w = 15$ C) $p = 130$ , $x = 0$ , $y = 15$ , $z = 0$ , $w = 15$ D) $p = 55$ , $x = 15$ , $y = 0$ , $z = 25$ , $w = 0$ E) no solution

Minimize $c = - x + 4 y$ subject to \[\begin{aligned} y & \leq \frac { 3 x } { 4 } \\ x & \leq 4 y \\ y & \geq 3 \\ x & \geq 4 \\ x + y & \leq 15 \end{aligned}\]

A) Min: $c = - 8$ , $x = 4$ , $y = 3$ B) Min: $c = 8$ , $x = 4$ , $y = 3$ C) Min: $c = 0$ , $x = 12$ , $y = 3$ D) Min: $c = 1$ , $x = 12$ , $y = 3$ E) Min: $c = - 0$ , $x = 4$ , $y = 3$ A) Min: $c = - 8$ , $x = 4$ , $y = 3$ B) Min: $c = 8$ , $x = 4$ , $y = 3$ C) Min: $c = 0$ , $x = 12$ , $y = 3$ D) Min: $c = 1$ , $x = 12$ , $y = 3$ E) Min: $c = - 0$ , $x = 4$ , $y = 3$

Solve the standard minimization problem using duality. Minimize $c = 0.7 s + 0.3 t$ Subject to \[\begin{array} { l } 30 s + 20 t \geq 600 \\ 0.1 s + 0.4 t \geq 4 \\ 0.2 s + 0.3 t \geq 4.5 \\ s \geq 0 . t \geq 0 \end{array}\]

A) $c = 9 , s = 30 , t = 0$ B) $c = 21 , s = 0 , t = 30$ C) $c = 30 , s = 30 , t = 30$ D) $c = 9 , s = 0 , t = 30$ E) $c = 21 , s = 30 , t = 0$ A) $c = 9 , s = 30 , t = 0$ B) $c = 21 , s = 0 , t = 30$ C) $c = 30 , s = 30 , t = 30$ D) $c = 9 , s = 0 , t = 30$ E) $c = 21 , s = 30 , t = 0$

Solve the standard minimization problem using duality. \[\begin{array} { l } \text { Minimize } & { c = 8 s + 7 t + 8 u } \\ &s + t + u \geq 100 \\ &2 s + t \geq 50 \\ \text { subject to } &t + u \geq 50 \\ &s \geq 0 , t \geq 0 , u \geq 0 \\ \end{array}\]

A) $c = 800$ , $s = 100$ , $t = 0$ , $u = 0$ B) $c = 800$ , $s = 0$ , $t = 0$ , $u = 100$ C) $c = 700$ , $s = 0$ , $t = 100$ , $u = 0$ D) $c = 350$ , $s = 0$ , $t = 50$ , $u = 0$ E) $c = 400$ , $s = 50$ , $t = 0$ , $u = 0$ A) $c = 800$ , $s = 100$ , $t = 0$ , $u = 0$ B) $c = 800$ , $s = 0$ , $t = 0$ , $u = 100$ C) $c = 700$ , $s = 0$ , $t = 100$ , $u = 0$ D) $c = 350$ , $s = 0$ , $t = 50$ , $u = 0$ E) $c = 400$ , $s = 50$ , $t = 0$ , $u = 0$

Minimize $c = 2 x + y + 3 z$ subject to \[\begin{aligned} x + y + z & \geq 50 \\ 2 x + y & \geq 10 \\ y + z & \geq 10 \\ x \geq 0 , y \geq 0 , z & \geq 0 \end{aligned}\] If there is no solution, do not fill the fields. $c =$ __________ $x =$ __________ $y =$ __________ $z =$ __________

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

Minimize $c = 3 x - 3 y$ subject to \[\begin{aligned} \frac { x } { 6 } & \leq y \\ y & \leq \frac { 2 x } { 3 } \\ x + y & \geq 5 \\ x + 3 y & \leq 20 \\ x \geq 0 , y & \geq 0 \end{aligned}\]

A) $c = 3$ , $x = 2$ , $y = 3$ B) $c = 6$ , $x = 2$ , $y = 3$ C) $c = 3$ , $x = 3$ , $y = 2$ D) the feasible region is empty E) the objective function is unbounded A) $c = 3$ , $x = 2$ , $y = 3$ B) $c = 6$ , $x = 2$ , $y = 3$ C) $c = 3$ , $x = 3$ , $y = 2$ D) the feasible region is empty E) the objective function is unbounded

You are planning to invest up to $11,600 in IBM, HPQ, and AAPL shares. You desire to maximize your share of the companies' earnings but, for tax reasons, want to earn no more than $ 230 in dividends. Your broker suggests that because AAPL stocks pay no dividends, you should invest only in AAPL. Is she right \[\begin{array} { | c | c | c | c | } \hline & \text { Price } & \begin{array} { c } \text { Dividend } \\ \text { Yield } \end{array} & \begin{array} { c } \text { Earnings } \\ \text { per share } \end{array} \\ \hline \text { IBM } & \$ 80 & 1 \% & \$ 5 \\ \hline \begin{array} { c } \text { HPQ (Hewlett } \\ \text { Packard) } \end{array} & \$ 20 & 2 \% & \$ 1 \\ \hline \text { AAPL (Apple) } & \$ 70 & 0 \% & \$ 1 \\ \hline \end{array}\]

The answer of You are planning to invest up to...

Write down (without solving) the dual LP problem. Minimize $c = s + 6 t + u$ subject to \[\begin{array} { l } 9 s - t + v \geq 3,000 \\ u - v \geq 1,000 \\ s + t \geq 500 \\ s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0 \end{array}\]

A) maximize \[p = x + 6 y + z\] subject to \[\begin{array} { l } 9 x - y + w \geq 3,000 \\ z - w \geq 1,000 \\ x + y \geq 500 \\ x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0 \end{array}\] B) maximize $p = 3,000 x + 1,000 y + 500 z$ subject to \[\begin{array} { l } 9 x + z \geq 1 \\ - x + z \geq 6 \\ y \geq 1 \\ x - y \geq 0 \\ x \geq 0 , y \geq 0 , z \geq 0 \end{array}\] C) maximize $p = 3,000 x + 1,000 y + 500 z$ subject to \[\begin{array} { l } 9 x + z \leq 1 \\ - x + z \leq 6 \\ y \leq 1 \\ x - y \leq 0 \\ x \geq 0 , y \geq 0 , z \geq 0 \end{array}\] D) maximize $p = x + 6 y + z$ subject to \[\begin{array} { l } 9 x - y + w \leq 3,000 \\ z - w \leq 1,000 \\ x + y \geq 500 \\ x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0 \end{array}\] E) maximize $p = x + 6 y + z$ subject to \[\begin{array} { l } 9 x + y + w \geq 3,000 \\ z + w \geq 1,000 \\ x + y \geq 500 \\ x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0 \end{array}\] A) maximize \[p = x + 6 y + z\] subject to \[\begin{array} { l } 9 x - y + w \geq 3,000 \\ z - w \geq 1,000 \\ x + y \geq 500 \\ x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0 \end{array}\] B) maximize $p = 3,000 x + 1,000 y + 500 z$ subject to \[\begin{array} { l } 9 x + z \geq 1 \\ - x + z \geq 6 \\ y \geq 1 \\ x - y \geq 0 \\ x \geq 0 , y \geq 0 , z \geq 0 \end{array}\] C) maximize $p = 3,000 x + 1,000 y + 500 z$ subject to \[\begin{array} { l } 9 x + z \leq 1 \\ - x + z \leq 6 \\ y \leq 1 \\ x - y \leq 0 \\ x \geq 0 , y \geq 0 , z \geq 0 \end{array}\] D) maximize $p = x + 6 y + z$ subject to \[\begin{array} { l } 9 x - y + w \leq 3,000 \\ z - w \leq 1,000 \\ x + y \geq 500 \\ x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0 \end{array}\] E) maximize $p = x + 6 y + z$ subject to \[\begin{array} { l } 9 x + y + w \geq 3,000 \\ z + w \geq 1,000 \\ x + y \geq 500 \\ x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0 \end{array}\]

Exam 6: Linear Programming

Minimize $c = 2.2 x + y + 3.3 z$ subject to x+1.5y+1.2z \geq100 2x+1.5y \geq40 1.5y+1.1z \geq40 x\geq0,y\geq0,z \geq0 Round answers to two decimal places, if necessary. If there is no solution, do not fill the fields. $c =$ $x =$ $y =$ $z =$

(Short Answer)

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Question 141

Maximize $p = 12 x + 10 y$ Subject to x+y \leq40 x \geq1 -x+2y \geq0 x\geq0,y \geq0

(Multiple Choice)

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Question 142

Maximize $p = 2 x + 3 y + z + 4 w$ Subject to x+y+z+w \leq40 2x+y-z-w \geq10 x+y+z+w \geq10 x\geq0,y\geq0,z\geq0,w \geq0

(Multiple Choice)

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Question 143

Minimize $c = - x + 4 y$ subject to y \leq x \leq4y y \geq3 x \geq4 x+y \leq15

(Multiple Choice)

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Question 144

Which inequality does the graph satisfy

(Multiple Choice)

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Question 145

Solve the standard minimization problem using duality. Minimize $c = 0.7 s + 0.3 t$ Subject to 30s+20t\geq600 0.1s+0.4t\geq4 0.2s+0.3t\geq4.5 s\geq0.t\geq0

(Multiple Choice)

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Question 146

Your farm encompasses 400 acres, and you are planning to grow soy beans, corn, and wheat in the coming planting season. Fertilizer costs per acre are: $5 for soy beans, $2 for corn, and $1 for wheat.You estimate that each acre of soy beans will require an average of 5 hours of labor per week, while tending to corn and wheat will each require an average of 2 hours per week. Based on past yields and current market prices, you estimate a profit of $3,000 for each acre of soy beans, $2,000 for each acre of corn, and $1,000 for each acre of wheat. You can afford to spend no more than $2,400 on fertilizer, and your farm laborers can supply 2,800 hours per week. How many acres of each crop should you plant to maximize total profits In this event, will you be using all the available labor ?

(Multiple Choice)

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Question 147

The Megabuck Hospital Corp. is to build a state-subsidized nursing home catering to homeless patients as well as high-income patients. State regulations require that every subsidized nursing home must house a minimum of 1,200 homeless patients and no more than 910 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 2,700 patients. The board of directors, under pressure from a neighborhood group, insists that the number of homeless patients should not exceed twice the number of high-income patients. Due to the state subsidy, the hospital will make an average profit of $10,000 per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at $9,000 per month. How many of each type of patient should it house in order to maximize profit ? If no optimal solution exists enter no solution in each box. ? Number of homeless patients = ? Number of high-income patients =

(Short Answer)

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Question 148

Solve the standard minimization problem using duality. Minimize c=8s+7t+8u s+t+u\geq100 2s+t\geq50 subject to t+u\geq50 s\geq0,t\geq0,u\geq0

(Multiple Choice)

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Question 149

Minimize $c = 2 x + y + 3 z$ subject to x+y+z \geq50 2x+y \geq10 y+z \geq10 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 150

Solve the standard minimization problem using duality. Minimize $c = 0.5 s + 0.1 t$ subject to 30s+20t\geq600 0.1s+0.4t\geq4 0.2s+0.3\geq4.5 s\geq0,t\geq0 $c =$ $s =$ $t =$ __

(Short Answer)

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Question 151

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 10 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 $52,500, 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 ? If no optimal solution exists enter no solution in each box. ? Number of spots on "Becker" = ? Number of spots on "The Simpsons." =

(Short Answer)

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Question 152

Minimize $c = 3 x - 3 y$ subject to \leqy y \leq x+y \geq5 x+3y \leq20 x\geq0,y \geq0

(Multiple Choice)

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Question 153

Maximize and minimize $p = 3 x - y$ subject to x+y\geq2 x-y\leq2 x-y\geq-2 x\leq12,y\leq12 Max: $p =$ , $x =$ , $y =$ Min: $p =$ , $x =$ , $y =$

(Short Answer)

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Question 154

Determine graphically the solution set for the system of inequalities. $\left\{ \begin{array} { c } 4 x - 12 y \leq 16 \\- x + 3 y \leq 4 \\x \geq 0 \\y \geq 0\end{array} \right.$

(Multiple Choice)

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Question 155

You are planning to invest up to $11,600 in IBM, HPQ, and AAPL shares. You desire to maximize your share of the companies' earnings but, for tax reasons, want to earn no more than $ 230 in dividends. Your broker suggests that because AAPL stocks pay no dividends, you should invest only in AAPL. Is she right Price Dividend Yield Earnings per share IBM \ 80 1\% \ 5 HPQ (Hewlett Packard) \ 20 2\% \ 1 AAPL (Apple) \ 70 0\% \ 1

(Short Answer)

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Question 156

Not every linear programming problem with a bounded feasible region has an optimal solution.

(True/False)

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Question 157

Graph the solution set to the inequality. Label coordinates of all corner points (if any). $- 7 x + 8 y \leq 24$

(Multiple Choice)

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Question 158

Write down (without solving) the dual LP problem. Minimize $c = s + 6 t + u$ subject to 9s-t+v\geq3,000 u-v\geq1,000 s+t\geq500 s\geq0,t\geq0,u\geq0,v\geq0

(Multiple Choice)

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Question 159

My friends: I, the mighty Brutus, have decided to prepare for retirement by instructing young warriors in the arts of battle and diplomacy. For each hour spent in battle instruction, I have decided to charge 90 ducats. For each hour in diplomacy instruction I shall charge 20 ducats. Due to my advancing years, I can spend no more than 60 hours per week instructing the youths, although the great Jove knows that they are sorely in need of instruction! Due to my fondness for physical pursuits, I have decided to spend no more than half of the total time in diplomatic instruction. However, the present border crisis with the Gauls is a sore indication of our poor abilities as diplomats. As a result, I have decided to spend 15 hours per week instructing in diplomacy. Finally, to complicate things further, there is the matter of Scarlet Brew: I have estimated that each hour of battle instruction will require at least 20 gallons of Scarlet Brew to quench my students' thirst, and that each hour of diplomacy instruction, being less physically demanding, requires half that amount. Because my harvest of red berries has far exceeded my expectations, I estimate that I'll have to use at least 1,350 gallons per week in order to avoid storing the fine brew at great expense. Given all these restrictions, how many hours per week should I spend in each type of instruction to maximize my income ?

(Multiple Choice)

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Question 160

Minimize $c = 2.2 x + y + 3.3 z$ subject to x+1.5y+1.2z \geq100 2x+1.5y \geq40 1.5y+1.1z \geq40 x\geq0,y\geq0,z \geq0 Round answers to two decimal places, if necessary. If there is no solution, do not fill the fields. $c =$ $x =$ $y =$ $z =$

Maximize $p = 12 x + 10 y$ Subject to x+y \leq40 x \geq1 -x+2y \geq0 x\geq0,y \geq0

Maximize $p = 2 x + 3 y + z + 4 w$ Subject to x+y+z+w \leq40 2x+y-z-w \geq10 x+y+z+w \geq10 x\geq0,y\geq0,z\geq0,w \geq0

Minimize $c = - x + 4 y$ subject to y \leq x \leq4y y \geq3 x \geq4 x+y \leq15

Which inequality does the graph satisfy

Solve the standard minimization problem using duality. Minimize $c = 0.7 s + 0.3 t$ Subject to 30s+20t\geq600 0.1s+0.4t\geq4 0.2s+0.3t\geq4.5 s\geq0.t\geq0

Solve the standard minimization problem using duality. Minimize c=8s+7t+8u s+t+u\geq100 2s+t\geq50 subject to t+u\geq50 s\geq0,t\geq0,u\geq0

Minimize $c = 2 x + y + 3 z$ subject to x+y+z \geq50 2x+y \geq10 y+z \geq10 x\geq0,y\geq0,z \geq0 If there is no solution, do not fill the fields. $c =$ $x =$ $y =$ $z =$

Solve the standard minimization problem using duality. Minimize $c = 0.5 s + 0.1 t$ subject to 30s+20t\geq600 0.1s+0.4t\geq4 0.2s+0.3\geq4.5 s\geq0,t\geq0 $c =$ $s =$ $t =$ __

Minimize $c = 3 x - 3 y$ subject to \leqy y \leq x+y \geq5 x+3y \leq20 x\geq0,y \geq0

Maximize and minimize $p = 3 x - y$ subject to x+y\geq2 x-y\leq2 x-y\geq-2 x\leq12,y\leq12 Max: $p =$ , $x =$ , $y =$ Min: $p =$ , $x =$ , $y =$

Determine graphically the solution set for the system of inequalities. $\left\{ \begin{array} { c } 4 x - 12 y \leq 16 \\- x + 3 y \leq 4 \\x \geq 0 \\y \geq 0\end{array} \right.$

Not every linear programming problem with a bounded feasible region has an optimal solution.

Graph the solution set to the inequality. Label coordinates of all corner points (if any). $- 7 x + 8 y \leq 24$

Write down (without solving) the dual LP problem. Minimize $c = s + 6 t + u$ subject to 9s-t+v\geq3,000 u-v\geq1,000 s+t\geq500 s\geq0,t\geq0,u\geq0,v\geq0

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Further Integration Techniques and Applications of the Integral

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Exam 6: Linear Programming

Maximize p=12x+10yp = 12 x + 10 yp=12x+10y Subject to x+y \leq40 x \geq1 -x+2y \geq0 x\geq0,y \geq0

Maximize p=2x+3y+z+4wp = 2 x + 3 y + z + 4 wp=2x+3y+z+4w Subject to x+y+z+w \leq40 2x+y-z-w \geq10 x+y+z+w \geq10 x\geq0,y\geq0,z\geq0,w \geq0

Minimize c=−x+4yc = - x + 4 yc=−x+4y subject to y \leq x \leq4y y \geq3 x \geq4 x+y \leq15

Which inequality does the graph satisfy

Solve the standard minimization problem using duality. Minimize c=0.7s+0.3tc = 0.7 s + 0.3 tc=0.7s+0.3t Subject to 30s+20t\geq600 0.1s+0.4t\geq4 0.2s+0.3t\geq4.5 s\geq0.t\geq0

Solve the standard minimization problem using duality. Minimize c=8s+7t+8u s+t+u\geq100 2s+t\geq50 subject to t+u\geq50 s\geq0,t\geq0,u\geq0

Minimize ​ c=2x+y+3zc = 2 x + y + 3 zc=2x+y+3z ​ subject to ​ x+y+z \geq50 2x+y \geq10 y+z \geq10 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= __________

Solve the standard minimization problem using duality. ​ Minimize c=0.5s+0.1tc = 0.5 s + 0.1 tc=0.5s+0.1t ​ subject to 30s+20t\geq600 0.1s+0.4t\geq4 0.2s+0.3\geq4.5 s\geq0,t\geq0 ​ c=c =c= __________ ​ s=s =s= __________ ​ t=t =t= __________

Minimize c=3x−3yc = 3 x - 3 yc=3x−3y subject to \leqy y \leq x+y \geq5 x+3y \leq20 x\geq0,y \geq0

Maximize and minimize p=3x−yp = 3 x - yp=3x−y subject to ​ x+y\geq2 x-y\leq2 x-y\geq-2 x\leq12,y\leq12 ​ Max: p=p =p= __________, x=x =x= __________, y=y =y= __________ ​ Min: p=p =p= __________, x=x =x= __________, y=y =y= __________

Determine graphically the solution set for the system of inequalities. {4x−12y≤16−x+3y≤4x≥0y≥0\left\{ \begin{array} { c } 4 x - 12 y \leq 16 \\- x + 3 y \leq 4 \\x \geq 0 \\y \geq 0\end{array} \right.⎩⎨⎧​4x−12y≤16−x+3y≤4x≥0y≥0​

Not every linear programming problem with a bounded feasible region has an optimal solution. ​

Graph the solution set to the inequality. Label coordinates of all corner points (if any). −7x+8y≤24- 7 x + 8 y \leq 24−7x+8y≤24

Write down (without solving) the dual LP problem. Minimize c=s+6t+uc = s + 6 t + uc=s+6t+u subject to 9s-t+v\geq3,000 u-v\geq1,000 s+t\geq500 s\geq0,t\geq0,u\geq0,v\geq0

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

Maximize $p = 12 x + 10 y$ Subject to x+y \leq40 x \geq1 -x+2y \geq0 x\geq0,y \geq0

Maximize $p = 2 x + 3 y + z + 4 w$ Subject to x+y+z+w \leq40 2x+y-z-w \geq10 x+y+z+w \geq10 x\geq0,y\geq0,z\geq0,w \geq0

Minimize $c = - x + 4 y$ subject to y \leq x \leq4y y \geq3 x \geq4 x+y \leq15

Solve the standard minimization problem using duality. Minimize $c = 0.7 s + 0.3 t$ Subject to 30s+20t\geq600 0.1s+0.4t\geq4 0.2s+0.3t\geq4.5 s\geq0.t\geq0

Minimize $c = 2 x + y + 3 z$ subject to x+y+z \geq50 2x+y \geq10 y+z \geq10 x\geq0,y\geq0,z \geq0 If there is no solution, do not fill the fields. $c =$ $x =$ $y =$ $z =$

Solve the standard minimization problem using duality. Minimize $c = 0.5 s + 0.1 t$ subject to 30s+20t\geq600 0.1s+0.4t\geq4 0.2s+0.3\geq4.5 s\geq0,t\geq0 $c =$ $s =$ $t =$ __

Minimize $c = 3 x - 3 y$ subject to \leqy y \leq x+y \geq5 x+3y \leq20 x\geq0,y \geq0

Maximize and minimize $p = 3 x - y$ subject to x+y\geq2 x-y\leq2 x-y\geq-2 x\leq12,y\leq12 Max: $p =$ , $x =$ , $y =$ Min: $p =$ , $x =$ , $y =$

Determine graphically the solution set for the system of inequalities. $\left\{ \begin{array} { c } 4 x - 12 y \leq 16 \\- x + 3 y \leq 4 \\x \geq 0 \\y \geq 0\end{array} \right.$

Not every linear programming problem with a bounded feasible region has an optimal solution.

Graph the solution set to the inequality. Label coordinates of all corner points (if any). $- 7 x + 8 y \leq 24$

Write down (without solving) the dual LP problem. Minimize $c = s + 6 t + u$ subject to 9s-t+v\geq3,000 u-v\geq1,000 s+t\geq500 s\geq0,t\geq0,u\geq0,v\geq0