CLASSROOM INVESTIGATION:

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1 MHS CLASSROOM INVESTIGATION: LINEAR PROGRAMMING LEARNING GOALS By completing this lesson you will: Practice establishing systems of inequalities Review graphing half-planes Investigate using a polygon of constraints to optimize a situational problem When working with linear programming problems we are given a large amount of information that we must sort through in order to optimize our solution. The most important part of the process is to carefully sort through the information and establish a system of inequalities.. Samira is opening a coffee shop and wants to have a combination of stools, which cost $8, and chairs, which cost $5. She needs at least 40 seats total, and she wants the number of chairs to be at most twice as many as the number of stools. There should be less than 30 chairs, and no more than 5 stools. Starting a business is expensive so she wants to find the best combination of seats to purchase that won t use much of her budget. Step : Define your variables. Remember, variables are the things that you don t know. Let x be the number of stools Let y be Step : Identify the important information in the problem. It will help to develop a system for organizing the key pieces so that you don t miss anything. Ex: underlining phrases like less than that have to do with inequalities, circling amounts in one unit (like km) and boxing amounts in another (like $), or highlighting information related to the same variable in the same colour. Classroom Investigation 83

2 Step 3: Establish your system of inequalities for Samira s coffee shop using the information you identified in Step. Helpful Hint: If variables can t be negative (like people or length), include x 0 and y 0 in your system of inequalities.. x + y Key Words: < less than > more than at most at least 5. x > 0 (Since she must have seats of each type) 6. Step 4: Determine the overall goal of the situation. This will be to either maximize or minimize something in the problem. The goal is to the (maximize/minimize) Step 5: Establish a rule to help you calculate the amount that you are maximizing/ minimizing. This rule is called the optimizing function. Here we are working with cost, so the rule will be C = *Note: All of the information you identified in Step should now be used in the inequalities and/or the optimizing function check to make sure nothing was left out Now that we have established our system of inequalities and optimizing function, the next step is to graph all of the inequalities (as half-planes) on one Cartesian plane. Do not graph the optimizing function we will come back to that later in the process. 84 Classroom Investigation

3 . Represent the system below on the graph: y < 3x + y x+ 4 y > x 3 x + 4y 8 *Rearrange inequalities to the form y = ax + b to graph with point-slope method Type of Line: < or > dotted or solid Shading: < or below > or above *Always test with a point before shading so you don t have to erase a half-plane Once the system of inequalities has been graphed, there will be an area where all of the half-planes overlap. We call this the polygon of constraints and the vertices of this polygon are where we will find the coordinates that achieve our optimization goal. *Tip: If it does not happen automatically, go back and shade your polygon slightly darker than the rest of the graph so it is easily identifiable. Classroom Investigation 85

4 3. Determine the coordinates of the vertices of the polygon of constraints below. With half-planes, solid lines are included in the solution and dotted lines are not, so we can only use vertices made up of intersecting solid lines. We can use vertices, and. Some of the points may fall on a crosshair and be determined by reading the graph. Here, we can see A(4, 5) and E(, ). Others can t be read as clearly, like D. Since D is at the intersection of y x+ and y x + 30, we will treat them like equalities (change the signs to =) and solve for the coordinates using the substitution, elimination, or comparison method. Show all your work to solve for the coordinates of point D. The possible solutions to this problem are points. When we ve established our possible solution points, we will use our optimizing function to see which coordinates achieve our goal. 86 Classroom Investigation

5 4. Determine which of the vertices (below) maximizes the optimizing function. P = 3x + 44y. Vertex Work Solution A (, 9) P = 3() + 44(9) = B (0, 6) C (8, 4) D (9, ) *Tip: Build yourself a table if you are not given one; it will help organize your answers The coordinates of the vertex that maximize the function are. Finally, after we ve determined the coordinates that optimize our goal we must make a final statement, putting our solution back into the context of the situation. 5. Going back to our original problem optimizing the seating situation at Samira s coffee shop, write a final statement for the solution (, 8). Since we established x to be the number of stools, the solution has stools, and since we established y to be the number of, the solution has. Therefore, our final statement would be: In order to Samira should purchase. Classroom Investigation 87