Exam 4: Inequalities and Linear Programming

Graph the solution of the system of inequalities. ​ ​

The simplex matrix shown below indicates that an optimal solution has been found ( when ) but that a second solution is possible. Find the second solution. ​ ​

Use the given feasible region determined by the constraint inequalities to find the maximum and minimum of the objective function . ​ ​

A contractor builds two types of homes. The Carolina requires one lot, $160,000 capital, and 160 worker-days of labor, whereas the Savannah requires one lot, $240,000 capital, and 160 worker-days of labor. The contractor owns 300 lots and has $48,000,000 available capital and 43,200 worker-days of labor. The profit on the Carolina is $40,000 and the profit on the Savannah is $52,000. Use the corner points of the feasible region to find how many of each type of home should be built to maximize profit. Find the maximum possible profit. Round your profit to the nearest dollar, another answers - to the nearest whole number. ​

A sausage company makes two different kinds of hot dogs, regular and beef. Each pound of beef hot dogs requires 0.75 lb of beef and 0.2 lb of spices, and each pound of regular hot dogs requires 0.17 lb of beef, 0.31 lb of pork, and 0.22 lb of spices. Suppliers can deliver at most 1020 lb of beef, at most 600 lb of pork, and at least 500 lb of spices. Let x represent the number of pounds of regular meat and y represent the number of pounds of beef. Write the inequalities that describe how many pounds of each type of hot dog can be produced. ​

A minimization problem is given. Solve its dual problem with the simplex method. ​ Minimize subject to ​ ​

Nolan Industries manufactures water filters/purifiers that attach to a kitchen faucet. Each purifier consists of a housing unit that attaches to the faucet and a 60-day filter that is inserted into the housing. Past records indicate that on average the number of filters produced plus the number of housing units produced should be at least 500 per week. It takes 20 minutes to make and assemble each filter and 40 minutes for each housing. The manufacturing facility has up to 20,000 minutes per week for making and assembling these units, but due to certain parts' supply constraints, the number of housing units per week can be at most 400. If manufacturing costs (for material and labor) are $7.25 for each filter and $8.55 for each housing unit, how many of each should be produced to minimize weekly costs? Find the minimum cost. ​

Use the simplex method to maximize the given function. Assume all variables are nonnegative. ​ Maximize subject to ​ ​

The graph of the feasible region is shown. Locate the corners of the feasible region in order to find the minimum of the given objective function (if it exists). ​ ​ ​

Assume that each nurse works 8 consecutive hours at the Beaver Medical Center. The center has the following staffing requirements for each 4-hour work period. ​ Work Period Nurses Needed If represents the number of nurses starting in period 1, the number starting in period 2, and so on, write and solve the linear programming problem that will minimize the total number of nurses needed. (Note that the nurses who begin work in period 6 work periods 6 and 1 for their 8-hour shift.) ​

Graph the solution of the system of inequalities. ​ ​

Use the simplex method to maximize the given function. Assume all variables are nonnegative. ​ Maximize subject to ​ ​

The Video Star Company makes two different models of DVD players, which are assembled on two different assembly lines. Line 1 can assemble 30 units of the Star model and 42 units of the Prostar model per hour, and Line 2 can assemble 150 units of the Star model and 40 units of the Prostar model per hour. The company needs to produce at least 250 units of the Star model and 210 units of the Prostar model to fill an order. Let x represent the number of assembly hours for Line 1 and y represent the number of assembly hours for Line 2. Graph the region determined by the constraint inequalities. ​

State the following problem in a form for which the simplex matrix can be formed (that is, as a maximization problem with constraints). ​ ​

Suppose a primal minimization problem and its dual maximization problem were solved by using the simplex method on the dual problem. The final simplex matrix is given. Find the solution of the maximization problem. Use x₁, x₂, x₃ as the variables and f as the function. ​ ​

A pork producer is considering two types of feed that contain the necessary ingredients for the nutritional requirements for fattening hogs. Brand X contains 2 units of ingredient A per pound and 15 units of ingredient B per pound, Brand Y contains 15 units of ingredient A per pound and 2 units of ingredient B per pound. The nutritional requirements for the hogs are at least 120 units of ingredient A and at least ​237 units of ingredient B. If brand X costs 65 cents per pound and brand Y costs 80 cents per pound, how many pounds of each brand should be bought to satisfy the nutritional requirements at minimum cost? ​

The graph of the feasible region is shown. Locate the corners of the feasible region in order to find the maximum of the given objective function (if it exists). ​ ​ ​

A primal maximization problem is given. Form the dual minimization problem. ​ Maximize subject to ​ ​

Graph the inequality. ​ ​

A woman has a building with 26 one-bedroom apartments, 40 two-bedroom apartments, and 60 three-bedroom apartments available to rent to students. She has set the rent at $550 per month for the one-bedroom units, $900 per month for the two-bedroom units, and $1,100 per month for the three-bedroom units. She must rent to one student per bedroom, and zoning laws limit her to at most 250 students in this building. There are enough students available to rent all the apartments. How many of each type of apartment should she rent to maximize her monthly revenue? Find the maximum possible monthly revenue. Round your profit to the nearest dollar. ​