mathematical modelling (calibration of dip stick)
DESCRIPTION
this is to introduce mathematical modelling in class rooms while teaching mathematicsTRANSCRIPT
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36 JNNSEC - 2009 SALT
LAKE STADIUM KOLKATAKOLKATA WEST BENGAL
th
M.S.R.PRASADM.S.R.PRASAD S.M.K.Z.P.H.SCHOOLMOVVAMOVVA
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SRI MANDAVA SRI MANDAVA KANAKAIAHKANAKAIAH
Z.P.H.SCHOOL
M O V V A
KRISHNA Dt. (A.P.)
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MATHEMATICAL
MODELLING
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WHAT IS MATHEMATICAL MODELLING?
REAL WORLD
PROBLEM
FORMULATION Mathematic
al
Problem
Mathematical
Solution
REAL WORLD
SOLUTIONINTERPRETATION
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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
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calibration of
DIP STICK
READY RECKONER
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zSPECIFYING THE
PROBLEMI O C
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PETROLEUM
SPECIFYING THE PROBLEM
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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.
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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.
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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)
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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
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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
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CROSS-SECTIONAL AREA OF OIL IN THE TANK
=180
( 1 - h ) sin A
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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
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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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