Study Guide Linear Programming Answer Key

STUDY GUIDE: LINEAR PROGRAMMING **#5-6: An objective function and a system of linear inequalities representing constraints are given. Graph the system of inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region. Use these values to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.**   1. A steel company produces two types of machine dies, part A and part B. The company makes a $2.00 profit on each part A that it produces and a $5.00 profit on each part B that it produces. Let x = the number of part A produced in a week and y = the number of part B produced in a week. Write the objective function that describes the total weekly profit.   2x + 5y = Profit 2. A dietician needs to purchase food for patients. She can purchase an ounce of chicken for $0.25 and an ounce of potatoes for $0.02. The dietician is bound by the following constraints:   * Each ounce of chicken contains 13 grams of protein and 24 grams of carbohydrates.   * Each ounce of potatoes contains 5 grams of protein and 35 grams of carbohydrates. The minimum daily requirements for the patients under the dietician's   care are 45 grams of protein and 55 grams of carbohydrates. Let x = the number of ounces of chicken and y = the number of ounces of potatoes purchased per patient. Write a system of inequalities that describes these constraints.   13x + 5y ≥ 45 24x + 35y ≥ 55 x ≥ 0 y ≥ 0 3. Mrs. White wants to crochet hats and afghans for a church fundraising bazaar. She needs 6 hours to make a hat and 4 hours to make an afghan, and she has no more than 56 hours available. She has material for no more than 12 items, and she wants to make at least   one afghan. Let x = the number of hats she makes and y = the number of afghans she makes. Write a system of inequalities that describes these constraints.   6x + 4y ≤ 56 x + y ≤ 12 y ≥ 1 x ≥ 0 y ≥ 0 **#7 & #8: A company manufactures two products X and Y. Each product has to be processed in three departments: welding, assembly, and painting. Each unit of X spends 2 hours in the welding department, 3 hours in assembly, and   1 hour in painting. The corresponding times for a unit of Y are 3   and 1 hours, respectively. The employer hours available in a month are 1,500 for the welding department, 1,500 in assembly, and 550 in painting. The contributions to profits are $100 for product X and $120 for product   Y.** 7. A) What is the objective function to be maximized in this linear programming problem, where Z is the total profit in $? Z = 100x + 120y B) What is the equation of the labor constraint line for the welding department? 2x + 3y ≤ 1500 hours 8. C) What is the equation of the labor constraint line for the assembly department? 3x + y ≤ 1500 hours D) What is the solution to this linear programming problem in terms of the respective quantities of X and Y to be produced if profits are to be maximized?   x = 400 y = 150 **Answer: A** 4. David and Harry operate a discount jewelry store. They want to determine the best mix of customers to serve each day. There are two types of customers for their store, retail (R) and wholesale (W). The cost to serve a retail customer is $70 and the cost to serve a wholesale   customer is $89. The average profit from either kind of customer is the same. To meet headquarters expectations, they must serve at least 8 retail customers and 12 wholesale customers daily. In addition, in order to cover their salaries, they must at least serve 30 customers each day. Which of the following is one of the constraints for this model?   R + W ≥ 30 5. Use the graph to the right to determine which of the following is NOT a solution to the system. [Image of a graph with a dotted line] (0, 12) (11, 11) (5, 15) (12, 0) **SYSTEMS OF EQUATIONS:** Determine whether the given ordered pair is a solution of the system. 11. (5, -4) ``` 3x + 5y = -5 10x - 10y = 100 ``` **Not a solution** 12. (1, 1) ``` x + y = 2 x - y = 0 ``` **Solution** Solve the system of equations by the substitution method. 13. x + y = 10 y = x - 2 **Solution: (6, 4)** 14. x + y = 6 y = 2x - 4 **No Solution** Solve the system by the addition method. 15. x + y = 7 x - y = 5 **Solution: (6, 1)** 16. 2x + 4y = 8 3x + 6y = 12 **Infinitely Many Solutions** 17. One number is 12 more than 7 times another number. The sum of the two numbers is 13. Find the two numbers. x = y + 12 x + y = 13 **Solution: x = 13, y = 1** 18. One number is 3 less than 4 times the first number. The sum of the two numbers is 11. Find the two numbers. x = 4y - 3 x + y = 11 **Solution: x = 7, y = 4**