Changes

== Concept 1 : Mathematical modelling ==

== Concept 1 : Mathematical modelling ==

One of the most important applications of mathematics in life is to find a optimal solution for problems. For example, what goods should a shopkeeper buy to get the most amount of profit or what is the optimal combination of food you should eat to get maximum amount of nutrients?   

One of the most important applications of mathematics in life is to find a optimal solution for problems. For example, what goods should a shopkeeper buy to get the most amount of profit or what is the optimal combination of food you should eat to get maximum amount of nutrients?   

<math>25x + 50y \leq 1000</math>

<math>25x + 50y \leq 1000</math>

Now we have four linear inequalities. There are multiple ways to solve these inequalities. The first method we will look at is the graphical method which is easier to visualize and understand.  

Now we have four linear inequalities. There are multiple ways to solve these inequalities. The method we will look here is the graphical method which is easier to visualize and understand.  

==== Activity ====

==== Activity ====

<math>25x + 50y \leq 1000</math>

<math>25x + 50y \leq 1000</math>

Geogebra file : https://www.geogebra.org/calculator/qvjg3ntd

{{Geogebra|shvcc7wh}}

The area in which all the colors are overlapping represents the region where all these inequalities are satisfied.  

The area in which all the colors are overlapping represents the region where all these inequalities are satisfied.  

==== Activity ====

==== Activity ====

Plot the area in which all the four inequalities overlap and plot all the vertices of the resulting polygon.  

Plot the area in which all the four inequalities overlap and plot all the vertices of the resulting polygon.  

Geogebra file : https://www.geogebra.org/calculator/qvjg3ntd

{{Geogebra|qvjg3ntd}}

 

Now we have four bounds of this polygon inside of which all the four inequalities are satisfied. All we need to do now is find a point in this set, where <math>x+y</math>is maximised. 

 

Simply looking at the the second plot, we can see that at A , <math>x+y</math>= 0, at C its 20, at D its 25 and finally at B its 30.  

 

This gives us our solution that '''''x = 20''''' and '''''y = 30''''' .  

[[File:Linear Programming plot 2 .png|center|frameless|700x700px]]

What this means in real life is that if the baker bakes 20 of the first type of cakes and 30 of the second type, they will end up with the most possible amount of cakes using 5kg flour and 1kg fat.  

==== Observation ====

== Recap ==

Now we have four bounds of this polygon inside of which all the four inequalities are satisfied. All we need to do now is find a point in this set, where <math>x+y</math>is maximised.

From above we can conclude that solving a optimisation problem using linear programming has following steps:

* Make a system linear inequalities using the constrains of the real life scenario. (For example we only had 5kg flour and 1kg fat and we used it to device the model)

* Plot the system of linear inequalities

* Plot the area which contains all the points that satisfy the inequalities.  

* Among those points, find the point which maximizes or minimizes the the variables (depending on your need). This point will always be one of the vertex of the polygon plotted by the linear equations. For example in our question, it was the vertex B(20,10) which maximized <math>x+y</math>