# Sketching And Modeling

A mathematical model is a mathematical description of a real life situation. The mathematical modeling is the process of creating a mathematical model, solving it and using it to understand the real life problem. The modeling has various steps involved as to understanding the model, solving it, interpreting it in the real life situation, and most importantly, validating the model.

## Modeling concepts:

The followings are some of the important concepts of the modeling.

1. To gain understanding:

If we have a mathematical model which reflects the essential behavior of a real world system of interest. It is the process of building the model we find out which factors are most important in the system.

2. To predict or forecast, or simulate:

In modeling the forecasting is very important in many types of organizations, since predictions of future events have to be incorporated into the decision making process.

3. To estimate:

For example, the contesting parties want to predict the probability of their party winning the elections. They want to estimate how many people in their constituency would vote for their party.

## Example and Sketch for Modeling:

The following exampe and sketch for modeling.

A man traveled 440 kilometers on 28 liters of petrol in a car. The man has to go by the car to a place which is 180 km away. How much petrol do they need?

Solution:

Step 1:Petrol needed for traveling 440 km = 28 liters

Petrol needed for traveling 180 km = ?

Mathematical description:

let x = distance he travel

y = petrol he need

y varies directly with x.

y = kx, where K is a constant

he can travel 440 kilometers with 28 liters of petrol.

Y = 28, x = 440

K = `y/x  = 28/440 = 1/` y / x = 28 / 440 = 1 / 10

Y = kx

Y = 1 / 10 x (1)

From this the equation can be describes the relationship between the petrol need and distance traveled.

Step 2:We want to find the petrol we need to travel 180 kilometers, so we have to find the value of y when x = 180. Putting x = 180 in (1), we have

Y = 180 / 10

= 18

Since, we need 18 liters of petrol to travel 180 kilometers.

Sketch for modeling:

The following figure shows the sketch of the modeling.