CCMIII U2D3 Warmup

Multiple Choice: Choose the best answer for each.  1. Solve x – (-38) ≥ -51 (a) x ≥ -89 (b) x ≤ -13 (c) x ≤ 89 (d) x ≥ -13  2. Solve 6x ˃ 5x + 19 (a) x ≤ 19 (b) x ≤ (c) x ≥ -19 (d) x ≥ 19  3. The sum of 2 consecutive odd integers is at most 12. What is the greatest integer?  (a) 3 (b) 5 (c) 7 (d) 9

-19

Distribute playing cards for Lego activity

Homework Check:

Document Camera

Homework Check:

Document Camera

Homework Check:

Document Camera

Homework Check:

Document Camera

M3U2D3 Linear Programming

OBJECTIVES:-Determine feasible regions pertaining to maximum and minimum conditions-Use geometric methods to solve multi – step problems

Linear Programming

Businesses use linear programming to find out how to maximize profit or minimize costs. Most have constraints on what they can use or buy.

The Lego Activity

The Lego Activity

# of tables# of chairs

16 10

The Lego Activity

2t c 12

2t 2c 16

2t + c < 12 2t + 2c < 16

t > 0 c > 0

The Lego Activity

The Lego Activity

4 Tables, 4 Chairs

Ex 1: Find the minimum and maximumvalue of the function f(x, y) = 3x - 2y.

We are given the constraints: y ≥ 2 1 ≤ x ≤5 y ≤ x + 3

Linear Programming

Find the minimum and maximum values by graphing the inequalities and finding the vertices of the polygon formed.

Substitute the vertices into the function and find the largest and smallest values.

6

4

2

2 3 4

3

1

1

5

5

7

8

y ≤ x + 3

y ≥ 2

1 ≤ x ≤5

Linear Programming

The vertices of the quadrilateral formed are:

(1, 2) (1, 4) (5, 2) (5, 8) Plug these points into the

original function: f(x, y) = 3x - 2y

Linear Programming

f(x, y) = 3x - 2y f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1 f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5 f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11 f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1

Linear Programming

f(1, 4) = -5 minimum f(5, 2) = 11 maximum

Ex 2:Find the minimum and maximum value of the function f(x, y) = 4x + 3y

We are given the constraints: y ≥ -x + 2 y ≤ x + 2 y ≥ 2x -5

1

4

6

4

2

53 4

5

1

1

2

3y ≥ -x + 2

y ≥ 2x -5

y x 1

24

Vertices

f(x, y) = 4x + 3y f(0, 2) = 4(0) + 3(2) = 6 f(4, 3) = 4(4) + 3(3) = 25 f( , - ) = 4( ) + 3(- ) = -1 = 7

3

1

3

1

3

7

3

28

3

25

3

Linear Programming

f(0, 2) = 6 minimum f(4, 3) = 25 maximum

Classwork

Corn & Beans

#1-3 checked in class

HW Check – Corn & Beans

Homework

Finish Corn & Beans

#4-7