**MATHS :: Lecture 14 :: Model**

**Definition **

**Model**

A mathematical model is a representation of a phenomena by means of mathematical equations. If the phenomena is growth, the corresponding model is called a growth model. Here we are going to study the following 3 models.

1. linear model

2. Exponential model

3. Power model

**1. Linear model **

The general form of a linear model is y = a+bx. Here both the variables x and y are of degree 1.

**To fit a linear model of the form y=a+bx to the given data.**

Here a and b are the parameters (or) constants of the model. Let (x1 , y1) (x2 , y2)…………. (xn , yn) be n pairs of observations. By plotting these points on an ordinary graph sheet, we get a collection of dots which is called a __scatter diagram.__

There are two types of linear models

(i) y = a+bx (with constant term)

(ii) y = bx (without constant term)

The graphs of the above models are given below :

‘a’ stands for the constant term which is the intercept made by the line on the y axis. When x =0, y =a ie ‘a’ is the intercept, ‘b’ stands for the slope of the line .

**Eg:1.** The table below gives the DMP(kgs) of a particular crop taken at different stages;

fit a linear growth model of the form w=a+bt, and find the value of a and b from the graph.

t (in days) ; |
0 |
5 |
10 |
20 |
25 |

DMP w: (kg/ha) |
2 |
5 |
8 |
14 |
17 |

**2. Exponential model **

This model is of the form y = aebx where a and b are constants to be determined

The graph of an exponential model is given below.

‘a’ stands for the constant term which is the intercept made by the line on the y axis. When x =0, y =a ie ‘a’ is the intercept, ‘b’ stands for the slope of the line .

**Eg:1.** The table below gives the DMP(kgs) of a particular crop taken at different stages;

fit a linear growth model of the form w=a+bt, and find the value of a and b from the graph.

t (in days) ; |
0 |
5 |
10 |
20 |
25 |

DMP w: (kg/ha) |
2 |
5 |
8 |
14 |
17 |

**2. Exponential model **

This model is of the form y = aebx where a and b are constants to be determined

The graph of an exponential model is given below.

o x

**Example:** Fit the power function for the following data

x |
0 |
1 |
2 |
3 |

y |
0 |
2 |
16 |
54 |

Crop Response models

The most commonly used crop response models are

- Quadratic model
- Square root model

Quadratic model

The general form of quadratic model is y = a + b x + c x2

The parabolic curve bends very sharply at the maximum or minimum points.

**Example**

Draw a curve of the form y = a + b x + c x2 using the following values of x and y

x |
0 |
1 |
2 |
4 |
5 |
6 |

y |
3 |
4 |
3 |
-5 |
-12 |
-21 |

Square root model

The standard form of the square root model is y = a +b+ cx

When c is negative the curve attains maximum

At the extreme points the curve bends at slower rate

** Three dimensional Analytical geometry**

Let OX ,OY & OZ be mutually perpendicular straight lines meeting at a point O. The extension of these lines OX1, OY1 and OZ1 divide the space at O into octants(eight). Here mutually perpendicular lines are called X, Y and Z co-ordinates axes and O is the origin. The point P (x, y, z) lies in space where x, y and z are called x, y and z coordinates respectively.

**Distance between two points **** **

The distance between two points A(x1,y1,z1) and B(x2,y2,z2) is

dist AB =

In particular the distance between the origin O (0,0,0) and a point P(x,y,z) is

OP =

**The internal and External section**

Suppose P(x1,y1,z1) and Q(x2,y2,z2) are two points in three dimensions.

P(x1,y1,z1) A(x, y, z) Q(x2,y2,z2)

The point A(x, y, z) that divides distance PQ internally in the ratio m1:m2 is given by

A = |

Similarly

P(x1,y1,z1) and Q(x2,y2,z2) are two points in three dimensions.

P(x1,y1,z1) Q(x2,y2,z2) A(x, y, z)

The point A(x, y, z) that divides distance PQ externally in the ratio m1:m2 is given by

A = |

If A(x, y, z) is the midpoint then the ratio is 1:1

A = |

**Problem**

Find the distance between the points P(1,2-1) & Q(3,2,1)

PQ= ===2

**Direction Cosines**

** **Let P(x, y, z) be any point and OP = r. Let a,b,g be the angle made by line OP with OX, OY & OZ. Then a,b,g are called the direction angles of the line OP. cos a, cos b, cos g are called the direction cosines (or dc’s) of the line OP and are denoted by the symbols I, m ,n.

**Result **

By projecting OP on OY, PM is perpendicular to y axis and the also OM = y

Similarly,

(i.e) *l = m = n = *

\*l2 + m2 + n2* =

(Distance from the origin)

\ l2 + m2 + n2 =

l2 + m2 + n2 = 1

(or) cos2a + cos2b + cos2g = 1.

**Note **

The direction cosines of the x axis are (1,0,0)

The direction cosines of the y axis are (0,1,0)

The direction cosines of the z axis are (0,0,1)

**Direction ratios**

Any quantities, which are proportional to the direction cosines of a line, are called direction ratios of that line. Direction ratios are denoted by a, b, c.

If l, m, n are direction cosines an a, b, c are direction ratios then

*a **µ** l, b **µ** m, c **µ** n*

(ie*) a = kl, b = km, c = kn*

(ie) (Constant)

(or) (Constant)

**To find direction cosines if direction ratios are given**

If a, b, c are the direction ratios then direction cosines are

*l = *

* similarly m = (1)*

* n = *

*l2+m2+n2 = *

(ie) 1 =

Taking square root on both sides

K =

\

**Problem**

1. Find the direction cosines of the line joining the point (2,3,6) & the origin.

**Solution **

By the distance formula

2. Direction ratios of a line are 3,4,12. Find direction cosines

**Solution**

Direction ratios are 3,4,12

(ie) a = 3, b = 4, c = 12

Direction cosines are

*l* =

* m*=

*n*=

**Note **

- The direction ratios of the line joining the two points A(x1, y1, z1) &

B (x2, y2, z2) are (x2 – x1, y2 – y1, z2 – z1) - The direction cosines of the line joining two points A (x1, y1, z1) &

B (x2, y2, z2) are

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