Exam 4: Inequalities and Linear Programming

A manufacturer makes Portable Satellite Radios and Auto Satellite Radios at plants in Lakeland and Rockledge. At the Lakeland plant, at most 1800 radios can be produced, and the production of the Auto Satellite Radios can be at most 200 fewer than the production of the Portable Satellite Radios. At the Rockledge plant, at most 1200 radios can be produced. The profits on the Portable Satellite Radios are $101 at Lakeland and $93 at Rockledge, and the profits on the Auto Satellite Radios are $71 at Lakeland and $76 at Rockledge. If the manufacturer gets a rush order for 1500 Portable Satellite Radios and 1300 Auto Satellite Radios, how many of each should be produced at each location so as to maximize profits? Find the maximum profit. ​

A cereal manufacturer makes two different kinds of cereal, Senior Citizen's Feast and Kids Go. Each pound of Senior Citizen's Feast requires 0.8 lb of wheat and 0.2 lb of vitamin-enriched syrup, and each pound of Kids Go requires 0.4 lb of wheat, 0.2 lb of sugar, and 0.2 lb of vitamin-enriched syrup. Suppliers can deliver at most 2880 lb of wheat, at most 850 lb of sugar, and at least 900 lb of the vitamin-enriched syrup. Let x represent the number of pounds of Senior Citizen's cereal and y represent the number of pounds of Kids Go cereal. Graph the region determined by the constraint inequalities. ​

A company manufactures two types of electric hedge trimmers, one of which is cordless. The cord-type trimmer requires 7 hours to make, and the cordless model requires 1.5 hours. The company has only 800 work hours to use in manufacturing each day, and the packaging department can package only 300 trimmers per day. Let x represent the number of cord type models and y represent the number of cordless models. Graph the solution of the system of inequalities. ​

Suppose that in a hospital ward, the patients can be grouped into two general categories depending on their condition and the amount of solid foods they require in their diet. A combination of two diets is used for solid foods because they supply essential nutrients for recovery. The table below summarizes the patient groups and their minimum daily requirements. Let x represent number of servings of Diet A and y represent number of servings of Diet B. Write the inequalities that describe how many servings of each diet are needed to provide the nutritional requirements. ​ ​

A farm co-op has 6,000 acres available to plant with corn and soybeans. Each acre of corn requires 9 gallons of fertilizer/herbicide and 0.75 hour of labor to harvest. Each acre of soybeans requires 3 gallons of fertilizer/herbicide and 1hour of labor to harvest. The co-op has available at most 40,508 gallons of fertilizer/herbicide and at most 5,250 hours of labor for harvesting. Find the maximum profit if the profits per acre are $65 for corn and $50 for soybeans. Round your answer to the nearest cent if necessary. ​

Patio Iron makes wrought iron outdoor dining tables, chairs, and stools. Each table uses 8 ft of a standard-width wrought iron, 2 hours of labor for cutting and assembly, and 2 hours of labor for detail and finishing work. Each chair uses 6 ft of the wrought iron, 2 hours of cutting and assembly labor, and 1.5 hours of detail and finishing labor. Each stool uses 1 ft of the wrought iron, 1.5 hours for cutting and assembly, and 0.5 hour for detail and finishing work, and the daily demand for stools is at most 16. Each day Patio Iron has available at most 108 ft of wrought iron, 50 hours for cutting and assembly, and 40 hours for detail and finishing. If the profits are $60 for each dining table, $49 for each chair, and $39 for each stool, how many of each item should be made each day to maximize profit? Find the maximum profit. Round your profit to the nearest dollar and another answers - to the nearest whole number. ​

Graph the inequality. ​ ​

Suppose a sausage company makes two different kinds of hot dogs, regular and all beef. Each pound of all-beef hot dogs requires 0.75 lb of beef and 0.2 lb of spices, and each pound of regular hot dogs requires 0.18 lb of beef, 0.3 lb of pork, and 0.2 lb of spices. Suppliers can deliver at most 982.5 lb of beef, at most 600 lb of pork, and at least 490 lb of spices. If the profit is $0.60 on each pound of all-beef hot dogs and $0.40 on each pound of regular hot dogs, how many pounds of each should be produced to obtain maximum profit? What is the maximum profit? ​

A simplex matrix is given. In this case, the solution is complete, so identify the maximum value of f and a set of values for the variables that gives this maximum value. If multiple solutions may exist, indicate this and locate the next pivot. ​ ​

Graph the solution of the system of inequalities. ​ ​

A simplex matrix for a standard maximization problem is given. Indicate whether or not the solution shown is complete (optimal). If the solution is not complete, find the next pivot or indicate that no solution exists. ​ ​

A political candidate wishes to use a combination of radio and TV advertisements in her campaign. Research has been shown that each 1-minute spot on TV reaches 0.6 million people and that each 1 -minute spot on radio reaches 0.4 million. The candidate feels she must reach 42 million people, and she must buy at least 54 minutes of advertisements. How many minutes of each medium should be used if TV costs $550 per minute, radio costs $110 per minute, and the candidate wishes to minimize costs? ​

A sausage company makes two different kinds of hot dogs, regular and all beef. Each pound of all-beef hot dogs requires 0.79 lb of beef and 0.24 lb of spices, and each pound of regular hot dogs requires 0.17 lb of beef, 0.34 lb of pork, and 0.22 lb of spices. Suppliers can deliver at most 1044 lb of beef, at most 402 lb of pork, and at least 300 lb of spices. Let x represent the number of pounds of regular meat and y represent the number of pounds of all beef. Graph the region determined by the constraint inequalities. ​

The Wellbuilt Company produces two types of wood chippers, economy and deluxe. The deluxe model requires 3 hours to assemble and 0.5 hour to paint, and the economy model requires 2 hours to assemble and 1 hour to paint. The maximum number of assembly hours available is 24 per day, and the maximum number of painting hours available is 8 per day. If the profit on the deluxe model is $18 per unit and the profit on the economy model is $14 per unit, find the maximum profit. Round your answer to the nearest dollar if necessary. ​

Solve the following linear programming problem. ​ Maximize subject to ​ ​

A sausage company makes two different kinds of hot dogs, regular and all beef. Each pound of all-beef hot dogs requires 0.75 lb of beef and 0.2 lb of spices, and each pound of regular hot dogs requires 0.18 lb of beef, 0.3 lb of pork, and 0.2 lb of spices. Suppliers can deliver at most 1,020 lb of beef, at most 600 lb of pork, and at least 500 lb of spices. If the profit is $0.70 on each pound of all-beef hot dogs and $0.40 on each pound of regular hot dogs, how many pounds of each should be produced to obtain maximum profit? What is the maximum profit? Round your profit to the nearest cent, another answers - to the nearest whole number. ​

Solve the following linear programming problem. ​ Minimize subject to ​ ​

A cereal manufacturer makes two different kinds of cereal, Senior Citizen's Feast and Kids Go. Each pound of Senior Citizen's Feast requires 0.6 lb of wheat and 0.2 lb of vitamin-enriched syrup, and each pound of Kids Go requires 0.4 lb of wheat, 0.2 lb of sugar, and 0.2 lb of vitamin-enriched syrup. Suppliers can deliver at most 2800 lb of wheat, at most 800 lb of sugar, and at least 1000 lb of the vitamin-enriched syrup. Let x represent the number of pounds of Senior Citizen's cereal and y represent the number of pounds of Kids Go cereal. Write the inequalities that describe how many pounds of each type of cereal can be made. ​

Use the simplex method to find the optimal solution. ​​ Minimize subject to ​

Suppose that three water purification facilities can handle at most 18 million gallons in a certain time period. Plant 1 leaves 20% of certain impurities, and costs $20,000 per million gallons. Plant 2 leaves 15% of these impurities and costs $30,000 per million gallons. Plant 3 leaves 10% impurities and costs $40,000 per million gallons. The desired level of impurities in the water from all three plants is at most 15%. If Plant 1 and Plant 3 combined must handle at least 14 million gallons, find the number of gallons each plant should handle so as to achieve the desired level of purity at minimum cost. Find the minimum cost. ​