💻 Models

This lesson covers core applied mathematics concepts and their agricultural applications for BSc Agriculture learners.

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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

1. Quadratic model
2. 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 = !lec14_clip_image044.gif

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

1. 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)
2. The direction cosines of the line joining two points A (x1, y1, z1) &

B (x2, y2, z2) are

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Summary Cheat Sheet

• Focus on core formulas, definitions, and solved patterns from this lesson.
• Practice stepwise derivations and numerical substitutions carefully.
• Connect each concept to practical agricultural problem-solving contexts.

References

1 source

Primary classroom notes and standard BSc Agriculture applied mathematics references.