36 JNNSEC - 2009 SALT

LAKE STADIUM KOLKATAKOLKATA WEST BENGAL

th

M.S.R.PRASADM.S.R.PRASAD S.M.K.Z.P.H.SCHOOLMOVVAMOVVA

SRI MANDAVA SRI MANDAVA KANAKAIAHKANAKAIAH

Z.P.H.SCHOOL

M O V V A

KRISHNA Dt. (A.P.)

MATHEMATICAL

MODELLING

WHAT IS MATHEMATICAL MODELLING?

REAL WORLD

PROBLEM

FORMULATION Mathematic

al

Problem

Mathematical

Solution

REAL WORLD

SOLUTIONINTERPRETATION

PROCESS OF MATHEMATICAL MODELLING

SPECIFYING THE REAL PROBLEM

SETTING UP A MATHEMATICAL MODEL

DETERMINING THE MATHEMATICAL PROBLEM

SOLVING THE MATHETICAL PROBLEM

INTERPRETING THE SOLUTION

USING THE SOLUTION OR IMPROVING THE MODEL

calibration of

DIP STICK

READY RECKONER

zSPECIFYING THE

PROBLEMI O C

PETROLEUM

SPECIFYING THE PROBLEM

Common Practice In Measuring The Amount Of Petrol Left In Underground Tank for STOCK CONTROL.

Some Type Of Dip Stick Is Used To Measure The HEIGHT Of Petrol In The Tank.

Basing on the Values of Heights,their

Corresponding Values of VOLUME are known

from some kind of conversion

tables&charts.

O

r-h

h

r

A B

D

E

SETTING UP A MODEL

ASSUMED CROSS-SECTIONAL AREA of

Underground petrol tank.

Volume will be proportional to the

cross-sectional area-not to the height

Relationship between the shaded area A,and the height h,provides a ready reckoner to convert height to area.

Formulating the Mathematical Problem

For simplicity let us take r =1m.

SOLVING the problem

E

O

B

r-hr=1 m

And then Cross- sectional area of Oil.

With in degrees

COS == r

==

r - h

In Triangle EOB

For values of h from 0 to 1.0 the angle is to be found out.

1 - h (Since r = 1 m)

A

O

E

Angle AOB = Ang.AOE +Ang.EOB

= 2

Area of Sector OAB = 2 360

X X2

(since r = 1m)

= 180

In AOB

Sq.m

1

B

CROSS-SECTIONAL AREA of OIL present in the tank

= Area of Sector OAB - Area of OAB

= A A

O

B

A

O

BE

O

180?

Area of OAB = Area of AOE + Area of EOB

= 2 X Area of AOE

= 2[ 12

]

= (1 - h) sin

( AOE = EOB) ~

x r sin x (1 -h )

B

CROSS-SECTIONAL AREA OF OIL IN THE TANK

=180

( 1 - h ) sin A

h(m) 0o A(m)0 0 0

0.1 25.84 0.0190.2 36.87 0.0520.30.40.50.6

2

0.70.80.9

1.0

0.0940.140.1960.2520.3120.3740.437

0.59084.2778.4772.5566.4260

53.14

45.59

Interpretation of the solution

Formulae used to develop the ready reckoner

COS = 1- h

To find for given h

A = 180

( 1- h ) Sin

Now we are in a position to design the

DIP STICK

HEIGHT

IS LINEAR

THE CROSS-SECTIONALAREA

IS NOT LINEAR

0

0.1

0.20.052

0

1.0

h(m)

0.5

A (Sq.m)

0.019

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.094

0.14

0.196

0.252

0.312

0.374

0.437

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L.TEJA SRI

SK.KHASIM BI

E.MADHU SRAVANI

D.BHUVANA