Students will work on each problem in groups. Each member of the group will complete the problem, but they must come to an agreement on the correct solution. Students will be given a grade as a group for each problem. Students will complete problem #1, turn in all papers stapled to the problem, and then pick up the next problem. There are four problems total.
Linear Programming Problem #1
A theater contains 500 seats. For an upcoming talent show, the theater manager plans to sell $4 and $5 tickets. He must sell at least 200 $4 tickets and 100 $5 tickets for the show to be produced, and he must bring in at least $2000 to break even. How many tickets at each price should be sold to maximize income? What is the maximum income?
__Objective Function__ __Variables__ __Constraints__
? ____________? x = # $4 tickets ? __________ ?
y = # $5 tickets y ≥ 100
? __________ ?
4x + 5y ≥ 2000
Linear Programming Problem # 2
Ernesto is about to take a history test consisting of matching questions worth 10 points each and essay questions worth 25 points each. He is required to do at least 3 matching questions, but time restricts him from doing more than 12. Similarly, he must do at least 4 essays, but time restricts him from doing more than 15. If Ernesto is required to answer a total of 20 questions, how many of each type should he answer to maximize his score? What is the maximum score?
__Objective Function__ __Variables__ __Constraints__
? ____________? x = matching x ≥ 3 , x ≤ 12
y = essay y ≥ 4 , y ≤ 15
? __________ ?
Linear Programming Problem # 3
A machine can produce either nuts or bolts, but not both at the same time. The machine can be used at most 8 hours a day. Furthermore, at most 6 hours a day can be used for making nuts and at most 5 hours a day can be used for making bolts. There is a $2 profit for each hour the machine makes nuts and a $3 profit for each hour the machine makes bolts. How many hours per day should the machine make each item in order to maximize profit? What is the maximum profit?
__Objective Function__ __Variables__ __Constraints__
? ____________ ? x = nut hours y ≤ 5
y = bolt hours x ≤ 6
? __________ ?
Linear Programming Problem #4
A carpentry shop makes dinner tables and coffee tables. Each week the shop must complete at least 9 dinner tables and 13 coffee tables to be shipped to furniture stores. The shop can produce at most 30 dinner tables and coffee tables combined each week. If the shop sells dinner tables for $120 and coffee tables for $150, how many of each item should be produced for a maximum weekly income? What is the maximum weekly income?
__Objective Function__ __Variables__ __Constraints__
____________ x = __________
y = __________
__________
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