Steps:1) Use this booklet and the Nayland maths website together to gain an understanding of the content.

2) BEFORE class, watch videos on the Nayland maths website of the lesson material & ideas, Make notes and try some examples.

3) DURING CLASS work through problems and examples.

4) Repeat steps 2) & 3)

5) When you feel ready try the Practice Assessment under ‘tests conditions’ in class

6) Mark & review the Practice Assessment

7) Do the Assessment

1) Find out what is required for Achieve, Merit & Excellence

2) Watch the Belgium Chocolate Maker’s videos.

http://maths.nayland.school.nz

3.2 Linear Programming

Don’t worry about the detail yet, just focus on the main idea

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Shading: if unsure test the point (0,0) Does it satisfy the inequality?Ex 12.01**Shade OUT

If you don’t know how to shade inequalities then watch these videos

Answers

{y≥0 .5x−2y<−3 x+4{2x+4 y≤8x−2 y≤6{ y<2y>2 x−2

{ y≥2 x−4y<−1.5 x+3{3 x−2 y≤122 x+ y≥6{y<−x+4

y>3x−4

{1.5 y+2x≥−122 .5 y−3x<9{ x+2 y≥4

3x−6 y≥−6{2 y+5 x<106 y−x>6

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{ y>0 .25 x+1 ¿} { y>−1¿ }¿{}{ y<−0 .5 x+2 ¿} { y>−3 ¿ }¿{}{ y>2 x−4 ¿ } {y>−2¿ }¿{}

{ y>0 .5x−2¿ } { y<−2 x+4 ¿ }¿ {}{ y>3 x−2 ¿ } {x>−1 ¿ }¿ {}{ y>2 x−5 ¿} { y<−1x+2¿ }¿{}

{ y≤3 x−5 ¿ }{ y≤3 ¿ }¿{}{3 y−2 x≤6 ¿ } { y≥−1 ¿ }¿{}{y≤−2 x−1y≥−x−3y≤0 .5x+2

{ y≥0 ¿ } {y≤4 ¿ } {x≥0 ¿ }¿{}{ y≤2 x−1 ¿ } { y≤2 ¿ }¿{}{5 y+2x≤10¿ } {y+x≥−1 ¿ }¿{}

Forming inequalities from mathematical sentenceseg. There are at least 4 ‘x’ for every ‘y’ So if y = 1 then x ≥ 4

if y = 2 then x ≥ 8 x ≥ 4y or y ≤ ¼x

eg. For each ‘P’ there are at most 6 ‘Q’ So if P = 1 then Q ≤ 6 if P = 2 then Q ≤ 12 Q ≤ 6P or P ≥ 1/6Q

Now try these ones:1) Altogether there are at least 10 ‘w’ and ‘h’ w + h ≥ 102) For every ‘a’ there are no more than 10 ‘b’

Ex 12.03

This is new, & a bit tricky so watch these videos first, before moving onto Ex 12.03

Finding max or min values for a given feasible region then do Ex 12.04

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a) Find the maximum value of 2x +3y, b) Find the maximum value of 5x + 4y given that x ≥ 0, y ≥ 0, x ≤ 4, y ≤ 5 & x + y ≤ 6 given the constraints x ≥ 0, y ≥ 0,

x + y ≤ 7, x + 2y ≤ 10

Ex 12.04

Worked answers to Ex 12.03 available on-line

There is a video of the solutions to these

problems

Worked answers to Ex 12.04 available on-line

STEPS: (LEARN THESE)1) Identify the objective function to be optimized and the constraints2) Form constraint equations eg Less than 5 staff are needed, At most 6 truck are available3) Graphically represent the constraints

These will be in the form of inequalities eg staff < 5, trucks ≤ 6Some constraints are implied eg staff ≥ 0 and trucks ≥ 0

4) Is the inequality line included? > < (not included so dotted line) vs ≥ ≤ (is included so solid line)5) Shaded the feasible region which fits all constraints (shade out is better)6) Find points of intersection between lines (Or points near if not whole numbers or < > (dotted lines)7) Test profit equation (Objective function) at points of ‘intersection’ to find maximum value

eg Profit = $10 000 × Trucks – $12 000 × Staff

Ex 12.06

Essential!

And this!

And these!

So I guess this is important...

1 2 3 4 5 6 7 8 1 2 3 4 5 6

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profit = y - x1 2 3 4 5 6 7 8 1 2 3 4 5 6

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6 Maximum profit

profit = y - 0.5x

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6 Range of optimal combinations

Multiple Solutions

If the objective function is linear (as it will be at level 3 stats) in the form: ax + byThis produces a value eg profit = k which is to be optimized ax + by = kRearranged so its in the form y=

This has a gradient of And y intercept of which is to be maximized

Changing ‘k’ doesn’t change the gradient of the objective function

Sliding the objective function to the limit of the feasible region to maximize ‘k’

If the objective function is parallel to a constraint line then multiple solutions are possible along that line

What values of ‘b’ in the objective function ax + by = k will still produce a maximum value at this point?The gradient will vary from -0.5 to infinite. The gradient of the objective function is

An airline offers economy & first class seats. An aircraft has 240m2 passenger space. Economy seats need 0.75m2 and first class 2m2. A maximum of 200 passengers allowed in the aircraft. Economy seats cost $400 and first class $7201) Find the maximum income?2) How many seats of each class give this maximum income?3) What values of the cost for first class are possible with the same number of seats producing the maximum income?

Remember the worked answers are available on-

line

Excellence stuff

y=−abx+ kb

−ab

kb

−ab

Ex 12.06

Popes Assignment 1

Popes Assignment 2

Worked answers for the assignments

below are available on-line

Popes Assignment 3

Popes Assignment 4

Now you are ready for the assessment