3.4 Pp
- 1. Bell Ringer: Graph the system of inequalities from problem # 36 on your homework last night. 2. Is (-1,1) a solution of the system? Why or why not?
- 2. Solve linear programming problems. Objective
- 3. optimization linear programming constraint feasible region New Vocabulary
- 4. Check It Out! Example 1 Graph the feasible region for the following constraints. (Hint: Find the vertices) x ≥ 0 y ≥ 1.5 2.5 x + 5 y ≤ 20 3 x + 2 y ≤ 12
- 5. Yum’s Bakery bakes two breads, A and B . One batch of A uses 5 pounds of oats and 3 pounds of flour. One batch of B uses 2 pounds of oats and 3 pounds of flour. The company has 180 pounds of oats and 135 pounds of flour available. Write the constraints for the problem and graph the feasible region. Example 1: Graphing a Feasible Region
- 6. Graph the feasible region.
- 7. The feasible region is a quadrilateral with vertices at (0, 0), (36, 0), (30, 15), and (0, 45). Check A point in the feasible region, such as (10, 10), satisfies all of the constraints.
- 8. Why would we want to find a feasible region? objective function:
- 11. Yum’s Bakery wants to maximize its profits from bread sales. One batch of A yields a profit of $40. One batch of B yields a profit of $30. Use the profit information and the data from Example 1 to find how many batches of each bread the bakery should bake. Example 2: Solving Linear Programming Problems
- 12. Example 2 Continued Step 1 Let P = the profit from the bread. Write the objective function: P = 40 x + 30 y Step 2 Recall the constraints and the graph from Example 1. x ≥ 0 y ≥ 0 5 x + 2 y ≤ 180 3 x + 3 y ≤ 135
- 13. Example 2 Continued Step 3 Evaluate the objective function at the vertices of the feasible region. Yum’s Bakery should make 30 batches of bread A and 15 batches of bread B to maximize the amount of profit. ( x , y ) 40 x + 30 y P($) (0, 0) 40(0) + 30(0) (0, 45) (30, 15) (36, 0)
- 14. Ticket out the Door Use your own words to define Vertex Principle of Linear Programming.
Editor's Notes
- #3: We have been solving linear inequalities. i.e. Word Problem from homework. I see that there are many solutions, but what is the BEST one?
- #4: Optimization: how to find the maximum or minimum of a quantity. Linear Programming: method of finding a maximum or minimum value (optimization) of a function given the constraints. Constraint : an INEQUALITY that restricts one of the variables Feasible Region : POSSIBLE SOLUTION
- #5: Have students graph on whiteboards and hold up. Intercepts: (8, 0) (0, 4) (4, 0), (6, 0) ** Must be careful with shading
- #6: Have students switch who has board and graph. DON’T FORGET TO COPY DOWN ALL IMPORTANT INFO FROM PROBLEM. Write inequalities on board.
- #7: Intercepts: (0, 90) (36, 0) Intercepts: (0, 45) (45, 0)
- #8: We need
- #9: Objective Function: The objective function may have a minimum, a maximum, neither, or both depending on the feasible region. **Typically a linear equation ** What is being maximized or minimized
- #12: Objective Function: The objective function may have a minimum, a maximum, neither, or both depending on the feasible region. **Typically a linear equation Why do you need the feasible region from Example 1 to solve the problem? Why do you think the vertices as well as points inside the feasible region can be used to evaluate the function? ** What is being maximized or minimized?
- #13: Recall: What does the Vertex Principle tell us? Set up chart
- #14: Call up students to fill in answers 0, 1350, 1650, 1440 WORKSHEET