September 10- September 15

Transcription

1 September 10- September 15

2 You will be given a sheet of paper to write your bell work on. If you need more room you may use an extra sheet of paper, but be sure to staple the scratch paper to the Bell Work The Bellwork/ Warmup sheet will be collected on Fridays and graded.

3 Monday Bellwork 1. Solve the system of equations 2. Multiply the matrices.

4 Warm- Up 1. How do you feel about this quiz? 2. Do you take pride in how you did on this quiz?

5 How to put it in your caclulator Go to Apps, arrow down to inequalz, enter, enter, enter Go to y= and put in your equations. To make the signs, arrow to the =, and 2 nd, Alpha, whichever sign you need To put in an equation for x, arrow up to x=, enter Hit graph Type in your inequality for x the same way 2 nd, Alpha, y= for shaded, 1, enter 2 nd, alpha, zoom for POI points of intersection, enter, arrow over to each point

6

7 What is the solution to this system?

8 In Linear Programming We look for the breaking even point: Where the profit me make on our merchandise is more than the cost to make it Where the two lines intersect

9 Solving Systems of Linear Inequalities Find the ordered pairs that satisfy both inequalities

10 First equation

11 Second Equation

12 Both Equations Therefore the solution to both inequalities would be the purple section of the graph ALL THE ORDERED PAIRS IN THE PURPLE SECTION ARE THE SOLUTIONS

13 Graph the inequality to solve Dominica is a graphic artist who makes greeting cards. Her startup cost will be $1500 plus $0.40 per card. In order for her to remain competitive with large companies, she must sell her cards for no more then $1.70 each. How many cards must she sell in order to make a profit.

14 Write the two equations to represent the situation y=.40x+1500 y= 1.70x X is the number of cards Y is the price

15 Graph the 2 Equations

16 What is the region in which she will make a profit.

17 Solve the Inequalities by graphing Example 1 x 0 y 0 2x + y 4 or y -2x+4

18 All Together Find the Vertices

19 (0,0) (2,0) (0,4)

20 Why do we need to find vertices? The vertices will be where the maximum and minimum values of the function lie (f(x,y)=ax+by+c). This is known as the Vertex Theorem.

21 VERTEX THEOREM The maximum or minimum value on a polygonal convex set occurs at a vertex of the polygonal boundary In other words. 1.Graph your inequalities 2.Find all Vertex points (ordered pairs) 3.Find the max and min by substituting them into the function.

22 Find the Maximum and Minimum Values of the Function For example, Lets look at the function f(x,y) = 5x - 3y with the following inequalities y 0 -x + y 2 0 x 5 x + y 6 1. Graph the inequalities (by hand or using your calculator) 2. Find the Vertices (by hand or calculator) 3. Substitute the values in the f(x,y) equation to find the maximum and minimum values). 4. Write your answer as. The maximum value of the function is at the point (, ) The minimum value of the function is at the point (, )

23 Find the Maximum and Minimum of the function f(x,y) = 5x - 3y

24 Vertices (0,0) (0,2) (2,4) (5,1) (5,0) Test the vertices in the equation to determine the maximum and minimum values. f(x,y) = 5x - 3y

25 Example 4: Find the maximum and minimum values of f(x,y) = x - y + 2 for the polygonal convex set determined by the system of inequalities. x + 4y 12 3x - 2y -6 x + y -2 3x - y 10

26 Level 1 Level 2 Practice Pick15 problems to do, at least 3 from each section. Please complete the problems by levels Level 3 Finding intesections of 3 or more lines Identify the feasible region of a system of inequalities Max/Minimize an objective function over a feasible region p109#5,13-15 p109#4,5,9-15 p109#6-7,17-22

27 2.7 Linear Programming Use Linear Programming procedures to solve applications Recognize Situations where exactly one solution to a linear programming application may not exist.

28 Linear Programming Many practical applications can be solved by using this method The nature of these problems is that certain constraints exist or are placed upon the variables, and some function of these variables must be maximized or minimized.

29 Procedure for Linear Programming 1. Define the variables x= y= 2. Write the constraints as a system of inequalities. 3. Graph the system 4. Find the vertices of the polygon formed 5. Write an expression to be minimized or maximized 6. Substitute values from the coordinates of the vertices into the expression 7. Select the greatest or least result.

30 Leading up to this section First we had to graph linear inequalities Then we learned to solve a system of inequalities Find the vertices Find the maximum and minimum values using substitution Now, we need to do all of this but also use our reasoning abilities to determine the function to be maximized or minimized and the constraints that form the region.

31 Example 1 Suppose a lumber mill can turn out 600 units of product each week. To meet the needs of its regular customers, the mill must prouduce 150 unit of lumber and 225 units of plywood. If the profit for each unit of lumber is $30 and the profit for each plywood is $45, how many units of each type of wood product should the mill produces to maximize profit?

32 Example 1 1. Write the variables X=unit of lumber produced y= units of plywood produced 2. Write the constraints as a system of inequalities x 150 there can not be less than 150 units of lumber y 225 there can not be less than 225 units of plywood produced x+y 600 The maximum number of units of product is 600 that the company can produce each week

33 Example 1 Graph the system Find the vertices (150,225) (375, 225) (150,450) Write the maximum and minimum function Since the profit of lumber is $30 per unit and the profit of plywood is $45 per unit, the profit function is P(x,y) = 30x+45y

34 Practice Level 4 Linear Programming Must: Pg. 111 #26-28; Pick 2 from pg. 115 #5-8,12-21