HN Math IIName______
Linear Programming Worksheet I
A) Write an expression for the quantity to be maximized or minimized(the objective function); B) Write all the constraints as a system of linear inequalities and graph; C) In a table, list the vertices (corner points) of the graph of the feasible region; D) List the corresponding values of the objective function for each of the vertices in the table; E) Answer the question asked in the problem.
1) The BackStage Dance Studio must plan for and operate many different classes, 7 days a week at all hours of the day. Saturday is a very important day for younger students, and each Saturday class fills up quickly. To plan the Saturday schedule, the director has to consider these facts: It’s not easy to find enough good teachers, so the studio can offer at most 8 tap classes and at most 5 jazz classes. The studio has limited classroom space, so it can offer at most 10 classes for the day. The studio makes a profit of $150 from each tap class and $250 from each jazz class. Find the schedule of classes that gives the maximum profit.
2) A carpenter makes bookcases in 2 sizes, small and large. It takes 4 hours to make a large bookcase and 2 hours to make a small one. The profit on the large bookcase is $35 and on a small bookcase it is $20. The carpenter can spend only 32 hours per week making bookcases and must make at least 2 of the large and at least 4 of the small each week. How many small and large bookcases should the carpenter make to maximize his profit? What is his profit?
3) Stitches Inc. can make at most 30 jean jackets and 20 leather jackets in a week. It takes 10 hours to make a jean jacket and 20 hours to make a leather jacket. The total number of hours by all of the employees can be no more than 500 hours per week. The profit on the jean jacket is $20, and the profit on a leather jacket is $50. How many of each type of jacket should be produced in order to maximize profit? What is the maximum profit?
4) Wheels Inc. makes mopeds and bicycles. Experience shows they must product at least 10 mopeds. The factory can product at most 60 mopeds and 120 bicycles per month. The profit on a moped is $134 and the profit on a bicycle is $20. If they can make at most 60 units combined, how many of each should the company make per month to maximize profit? What is the maximum profit?
HN Math IIName______
Linear Programming Worksheet II
1) A sporting goods manufacturer produces skateboards and in-line skates. Its dealers demand at least 30 skateboards per day and 20 pairs of in-line skates per day. The factory can make at most 60 skateboards and 40 pairs of in-line skates per day. The total number of skateboards and pairs of in-line skates cannot exceed 90. The profit on each skateboard is $12 and the profit on each pair of in-line skates is $18. How many of each product should the company manufacture to get the maximum profit? What is the profit?
2) A bakery is making whole-wheat bread and apple bran muffins. For each batch of bread, they make $35 profit. For each batch of muffins, they make $10 profit. The bread takes 4 hours to prepare and 1 hour to bake. The muffins take .5 hours to prepare and .5 hours to bake. The maximum preparation time available is 16 hours and the maximum bake time is 10 hours. How many batches of bread and muffins should be made to maximize profits? What is the maximum profit?
3) When architects design buildings, they have to balance many factors. Construction and operating costs, strength, ease of use, and style of design are only a few. For example, when architects designing a large city office building began their design work, they had to deal with the following conditions:
a) The front of the building had to use windows of traditional style to fit in with the surrounding historic
buildings. There had to be at least 80 traditional windows on the front of the building. Those
windows each had an area of 20 square feet and glass that was 0.25 inches thick.
b) The back of the building was to use modern style windows that had an area of 35 square feet and glass
that was 0.5 inches thick. There had to be at least 60 of those windows.
c) In order to provide as much natural lighting for the building as possible, the design had to use at least
150 windows.
d) The traditional windows cost $200 each, and the modern windows cost $250 each.
If you were the architect, what combination of traditional and modern windows would you recommend to minimize total cost? What would be the minimum cost?
HN Math IIName______
Linear Programming Worksheet III
1) Paisan’s Pizza makes gourmet frozen pizzas for sale to supermarket chains. They make only deluxe pizzas, one vegetarian and the other with meat. Their business planning has these constraints and objective: Each vegetarian pizza takes 12 minutes of labor and each meat pizza takes 6 minutes of labor. The plant has at most 3,600 minutes of labor available each day. The plant freezer can handle a total of at most 500 pizzas per day. Vegetarian pizza is not quite as popular as meat pizza, so the plant makes at most 200 of this type each day. The sale of each vegetarian pizza earns Paisan’s $3 profit and each meat pizza earns $2 profit. Find the number of each type of pizza that will maximize profit and find the maximum profit.
2) The manufacturing facility that supplies a chain of Packaging Plus stores received a rush order for 290 boxes. It had to fill the order in eight hours or less. The factory has a machine that can produce 30 boxes per hour and costs $15 per hour to operate. The factory can also use two student workers from other, less-urgent tasks; together those students can make 25 boxes per hour at a cost of $10 per hour. What combinations of machine and student work times will meet the order deadline for least total cost?
3) TV Electronics Inc. makes console and wide-screen televisions. The equipment in the factory allows for manufacturing at most 450 console televisions and 200 wide-screen televisions in one month. The chart below shows the cost of making each type of television, as well as the profit for each.
Television / Cost Per Unit / Profit Per UnitConsole / $600 / $125
Wide-Screen / $900 / $200
How many of each type should be produced in order to maximize profit? What is the maximum profit?
4) For astronauts, the selection of a good diet is a carefully planned scientific process. Each person wants a high performance diet for a minimal total weight. Consider the following simplified version of a problem facing NASA space flight planners who must provide food for astronauts: There are two kinds of food to be carried on a space shuttle trip: special food bars or cartons of a special drink. Each food bar provides 5 grams of fat, 40 grams of carbohydrates, and 8 grams of protein. Each drink provides 6 grams of fat, 25 grams of carbohydrates, and 15 grams of protein. Minimum daily requirements for each astronaut are at least 61 grams of fat, at least 350 grams of carbohydrates, and at least 103 grams of protein. Each food bar weighs 65 grams and each carton of drink weighs 118 grams. Determine what combination of food bars and drinks will give the minimum daily requirements of fat, carbohydrates, and protein with the least total weight.