3.4 Notes: Systems of Linear Inequalities Name Introduction to Linear Programming PAP Alg II

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1 3.4 Notes: Sstems of Linear Inequalities Name Introduction to Linear Programming PAP Alg II Date Per Vocabular sstem of inequalities that bounds the shaded or feasible region; can also be called restrictions shaded area bounded b the constraints a function that describes the goal of the problem boundar point of the feasible region ielding the largest solution of the objective function boundar point of the feasible region ielding the smallest solution of the objective function 1. Using linear programming, find the values of and that maimize and minimize the objective function P = Restrictions Step 1 Step 2 Step 3 Graph the restrictions Find the coordinates of each Evaluate P at each verte Verte of the region P = 10 15

2 Use linear programming. Find the values of and that maimize and minimize each objective function P = P = P = - 2

3 3.4 WS: Sstems of Linear Ineq. & Intro to Linear Programming Pre-AP Algebra II Name: Graph each sstem of inequalities > 12 8 > 4 2. < 5 2 > 0 5 > < Solve the following sstems graphicall. Indicate whether the solution region is bounded or unbounded. List the coordinates of each corner point (verte) Vertices: Bounded/Unbounded? Vertices: Bounded/Unbounded?

4 Vertices: Bounded/Unbounded? Use linear programming. Find the values of and that maimize and minimize each objective function P = P = - 2

5 3.5 Notes: Linear Programming Name Pre-AP Algebra II Date Per The Supreme Shipping Compan can load its trucks with both rectangular and clindrical containers. A rectangular container has a volume of 100 cubic ft and weighs 200 lb. A clindrical container has a volume of 200 cubic feet and weighs 100 lb. Let denote the number of rectangular crates carried b a truck, and let denote the number of clindrical containers. 1. What constraint must be satisfied if each truck has room for at most 4200 cubic ft of containers? 2. What constraint must be satisfied if each truck can carr a maimum of 4800 lb? 3. What additional constraints must be satisfied because the problem involves real objects? 4. Graph the feasibilit set on the grid and label its vertices. Call the verte on the -ais A, the verte on the -ais B, and the verte on neither ais C. 5. Suppose that Supreme Shipping charges $50 to ship a rectangular and $60 to ship a clindrical container and wishes to maimize its income. a. What is the objective function? b. What is the value of the objective function at verte A? c. At verte B? d. At verte C? e. What combination of containers should Supreme Shipping use to maimize its income? What would be the maimum income?

6 Problem Solving: Linear Programming 6. Each quart of the regular skin lotion that Jason makes contains 2 cups of oil and 1 cup of cocoa butter. Each quart of his etra rich lotion contains 1 cup of oil and 2 cups of cocoa butter. Jason makes $10 profit on each quart of the regular lotion and an $8 profit on each quart of the etra rich lotion. If he has 12 cups of oil and 8 cups of cocoa butter on hand, how man quarts of each tpe of lotion should he make to maimize his profits? What would his profit be? 7. A loaf of Irish soda bread requires 4 cups of flour and 1 cup of sugar. A loaf of zucchini bread uses 2 cups of flour and 1 cup of sugar. Mari Alice has 16 cups of flour and 7 cups of sugar on hand. She makes $2 profit per loaf of Irish soda bread and $3 per loaf of zucchini bread. To maimize profits, how man loaves of each tpe should she make if she must make some of each? What would her profit be?

7 3.5 WS Linear Programming Name Pre-AP Algebra II 1. A manufacturer produces to planes and cars and her machines operate 24 hours a da. To produce a to plane requires 2 hours of work b machine A and 6 hours on machine B. To produce a to car requires 6 hours on machine A and 2 hours on machine B. The manufacturer earns a profit of $12 on each plane and $9 on each car. How man of each should be produced to maimize her profit? What would her profit be? 2. Best Video has budgeted $1800 per month for advertising. It will place ads with the two local radio stations with the largest teenage market share. Each 30 second spot on KJAM will cost $200 and will reach 15,000 teenage listeners. Those on KGRV will cost $300 and will reach 25,000 teenagers. KGRV has at most 4 spots available and KJAM has at most 6. How man commercial spots should be aired on each station to maimize the number of teenagers reached? How man teenagers would be reached?

8 3. A compan makes two kinds of tires: model R and model S. Each tire is processed on three machines: A, B, and C. To make one model R requires 1/2 hour on machine A, 2 hours on machine B, and 1 hour on C. To make one model S requires 1 hour on A, 1 hour on B, and 4 hours on C. During the net week machine A will be available for at most 20 hours, machine B for at most 60 hours, and machine C for at most 60 hours. If the compan makes a $10 profit on each model R tire and $15 profit on each model S tire, about how man of each tire should be made to maimize the compan s profit? What would the profit be? 4. Maddo Tours provides tour guide services and charter buses or vans according to demand. A bus carries 44 tourists and 3 guides and costs Maddo $800 per da. A van carries 11 tourists and 1 guide and costs Maddo $70 per da. A convention has indicated that a minimum of 396 delegates are epected to register for tours. Maddo has a maimum of 30 guides available. What mi of buses and vans should be used to minimize costs? What would be the minimum cost?