Exam 6: Linear Programming

Select the correct graph of the given system of inequalities and indicate whether the solution set is bounded or unbounded. $\left\{ \begin{array} { c } 7 x + 5 y \leq 25 \\5 x + y \geq 5 \\x + y \geq 3 \\x \geq 0 , y \geq 0\end{array} \right.$

Maximize $p = 4 x + 9 y$ subject to the following (using the simplex method). ​ 8x+5y \leq80 x+y \geq5 5x+8y \leq80 x\geq0,y \geq0 ​ $p =$ __________ ​ $x =$ __________ ​ $y =$ __________

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 600 small cars, at least 500 medium cars, and at least 600 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible What will be the total cost ?

Your small farm encompasses 100 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: $5 for tomatoes, $4 for lettuce, and $2 for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of $2,000 for each acre of tomatoes, $1,500 for each acre of lettuce and $500 for each acre of carrots. You would like to spend at least $300 on fertilizer (your niece owns the company that manufactures it) and your farm laborers can supply up to 500 hours per week. How many acres of each crop should you plant to maximize total profits In this event, will you be using all 100 acres of your farm ?

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 700 homeless patients and no more than 620 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 1,800 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 $13,000 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 ?

Your friend's portfolio manager has suggested two high-yielding stocks: Consolidated Edison (ED) and Royal Bank of Scotland (RBS-K). ED shares cost $40 and yield 5.5% in dividends. RBS-K shares cost $25 and yield 7.5% in dividends. You have up to $25,200 to invest, and would like to earn at least $1,386 in dividends. Draw the feasible region that shows how many shares in each company you can buy. Find the corner points of the region. (Round each coordinate to the nearest whole number.) Let the number of shares in ED be x, and the number of shares in RBS-K be y.

Oz makes lion food out of giraffe and gazelle meat. Giraffe meat has 18 grams of protein and 36 grams of fat per pound, while gazelle meat has 36 grams of protein and 18 grams of fat per pound. A batch of lion food must contain at least 36,000 grams of protein and 54,000 grams of fat. Giraffe meat costs $4/pound and gazelle meat costs $8/pound. How many pounds of each should go into each batch of lion food in order to minimize costs What are the shadow costs of protein and fat Use the rounded numbers of pounds in your calculations. ?

Graph the solution set to the inequality. Label coordinates of all corner points (if any). $\left\{ \begin{array} { l } x + y \leq 3 \\2 x + y \geq 6 \\y \geq 0\end{array} \right.$

Minimize ​ $c = 50 x + 11 y + 50 z$ ​ subject to ​ 3x+z \geq4 3x+y-z \geq3 4x+y-z \leq4 x\geq0,y\geq0,z \geq0 ​ If there is no solution, do not fill the fields. ​ $c =$ __________ ​ $x =$ __________ ​ $y =$ __________ ​ $z =$ __________

The Enormous State University's Business School is buying computers. The school has two models to choose from. Each first model comes with 400 MB of memory and 80 GB of disk space, while each second model has 300 MB of memory and 100 GB of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least 48,000 MB of memory and at least 12,800 GB of disk space. If both models cost $800 each, how many of each should the school buy to keep the cost as low as possible __________ first model(s) and __________ second model(s). What are the shadow costs of memory and disk space Please, enter first the shadow cost of MB and then the shadow cost of GB as improper fractions of integers without units. Separate your answers by a comma.

Solve the games with the given payoff matrix. $P = \left[ \begin{array} { l l l } - 4 & - 6 & - 3 \\- 4 & - 3 & - 5\end{array} \right]$

Given the set of constraints ? $\left\{ \begin{array} { c } 9 x + 6 y \leq 36 \\2 x + y \leq 7 \\x \geq 0 , y \geq 0\end{array} \right\}$ ? where does the maximum of $p = 3 x + 2 y$ occur at ? Please enter your answer as an ordered pair, if there is more than one ordered pair, separate your answers with commas.

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 2,600. 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 ?

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

Enormous State University's Business School is buying computers. The school has two models to choose from, the Pomegranate and the iZac. Each Pomegranate comes with 400 MB of memory and 80 GB of disk space, and each iZac has 300 MB of memory and 100 GB of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least 18,000 MB of memory and at least 4,400 GB of disk space. Draw the feasible region that shows the number of each kind of computer it can buy. Find the corner points of the region. Let the number of Pomegranate models be x, and the number of iZac models be y.

Use the simplex method to maximize $p = 3 x + 8 y + z$ subject to the following. ​ 6x+2y+2z \leq30 x+3y+4z \leq6 3x+5y+5z \leq14 $x \geq 0 , y \geq 0 , z \geq 0$ ​ $p =$ __________ ​ $x =$ __________ ​ $y =$ __________ ​ $z =$ __________

Solve the standard minimization problem using duality. Minimize $c = 5 s + 2 t$ Subject to 3s+t\geq30 s+t\geq20 s+3t\geq30 s\geq0,t\geq0

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and no Vitamin C. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 45 percent of the U.S. Recommended Daily Allowance (RDA) of Vitamin C for infants. Each serving of Gerber Apple Banana Juice contains 60 calories and 120 percent of the RDA of Vitamin C for infants. The cereal costs 10 cents per serving, the dessert costs 53 cents per serving, and the juice costs 27 cents per serving. If you want to provide your child with at least 360 calories and at least 240 percent of the RDA of Vitamin C, how can you do so at the least cost ? __________ serving(s) of cereal, __________ serving(s) of dessert, and __________ serving(s) of juice.

Your publishing company is about to start a promotional blitz for its new book, Physics for the Liberal Arts. You have 35 salespeople stationed in Chicago and 10 in Denver. You would like to fly at most 20 into Los Angeles and at most 20 into New York. A round-trip plane flight from Chicago to LA costs $195; from Chicago to NY costs $182; from Denver to LA costs $395; and from Denver to NY costs $166. You want to spend at most $7,380 on plane flights. How many salespeople should you fly from each of Chicago and Denver to each of LA and NY to have the most salespeople on the road ?

Graph the solution set to the inequality. Label coordinates of all corner points (if any). $x \geq - 6 y$