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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