Céline Mortier and Karen Sijmons, 6 ECMT
Catharina Stappers, Abigail Van den Eynde, Dorien Van Poucke, 6 MW
CONTENTS
1. 7step plan : Survey
2. Description of the experiment
3. Materials used
4. Measurements observed
5. Simpson’s formula: proof
6. Calculations : applied on a beer glass “Westmalle”
7. Control
8. History
1. 7STEP
PLAN : SURVEY
Need:
determination of the content of a “Westmalle” beerglass
determination of the function(of the shape) of the beerglass and drawing the graph
Problem:
we need to determine the measuring points as accurately as
possible.
the possibilities concerning the choice of the instruments is
limited.
Possible solutions:
by means of a vernier calliper gauge
by means of a micrometer to measure the inner diameter of cylinders
by means of a micrometer to measure the inflection of a rotating
axis up to 0,01 mm
by means of a pair of compasses
Executing one of the possible solutions:
measuring the points by means of a pair of compasses.
Evaluation:
it is rather cumbersome because we don’t have any fixed adjustable
material at
our disposal
which is very accurate
Possibilities that weren’t suitable:
vernier
calliper gauge : the jaws weren’t long enough for the depth of the glass.
micrometer : the curve of the glass became too slanting.
Control of the results:
pouring a liquid(f.e. water) into the glass until it is completely
filled. Afterwards the liquid is
poured in a measuring glass and this measure is considered as a control(1).
applying another method to calculate the content: the rule of Bezout.
2. DESCRIPTION OF THE EXPERIMENT
We start with the determination of the depth of the
glass. We do this by means of a vernier
calliper gauge and a ruler that is placed right across the glass. We obtain the following
measurements : 81.50 mm – 0,50 mm(ruler) = 81,00 mm.
This depth has to be divided into an odd number of equal intervals(height of the measuring
points). We make a stick long enough to equal the depth of the glass plus the length of the legs
of the compass. On the stick we apply a scale division : we put down
a mark every 9,00 mm.
After each measurement we saw off 9,00 mm to be able to determine the next point. We
continue to do so until all the marks have been sawn off and so we finally reach the number of
10 measuring points (0 included) and 9 intervals.
To work even more accurately we fill the glass with water. In this way we have a level measuring
surface for each mark and this means we can be sure to measure the diameter at the right height.
We have taken the thickness of the glass into account for we have measured the points on the
inside of the glass.
3. MATERIALS USED
wood saw, wood file, syringe, stick of wood
vernier calliper gauge and pair of compasses
4. MEASUREMENTS OBSERVED

diameter(cm) 
height(cm) 
radius(cm) 
surface(cm²) 
Measurement 1 
10.550 
8.100 
5.275 
87.410 
mean 
10,475 

5.237 
86.175 
Measurement 2 
10.400 
7.200 
5.200 
84.940 
mean 


5.161 
83.310 
Measurement 3 
10.245 
6.300 
5.122 
81.710 
mean 


5.061 
80.120 
Measurement 4 
10.000 
5.400 
5.000 
78.530 
mean 


4.875 
74,705 
Measurement 5 
9.500 
4.500 
4.750 
70.880 
mean 


4.637 
67,605 
Measurement 6 
9.050 
3.600 
4.525 
64.330 
mean 


4.337 
59.220 
Measurement 7 
8.300 
2.700 
4.150 
54.110 
mean 


3.940 
48.910 
Measurement 8 
7.460 
1.800 
3.730 
43.710 
mean 


3.400 
36.660 
Measurement 9 
6.140 
0.900 
3.070 
29.610 
mean 


2.185 
17.460 
Measurement 10 
2.600 
0.000 
1.300 
5.310 
Céline Mortier en Karen Sijmons
5. SIMPSON’S FORMULA : PROOF
We will prove, by means of a simple example, that the formula of Simpson, which we have used
to measure the content of the beer glass, is right.
*Surface under oblique line = surface of a rectangle + surface of a rectangular triangle
= base . height + base. height . ½
= 4 X 4 + 4 X 4 X ½
= 24 (in cm²)
*Simpson’s rule in general :
1/3 X L (y0 + 4.y1 + 2.y2 + … 2.yn2 + 4.yn1 + yn)
whereby L is the interval
Condition for applying the formula of Simpson is :there must be an odd number of
ycoordinates.
*For the given example the surface is:
1/3 X 1cm X (1 X 4cm + 4 X 5cm + 2 X 6cm + 4 X 7cm + 1 X 8cm )
= 72/12 cm² = 24 cm²
6.CALCULATIONS : applied on a beer glass “Westmalle”
1. What is the content of a full glass?
We have 10 measurements: the volume of the upper slice plus the volume calculated on the basis
of Simpson’s formula equals the volume of the whole glass.
Interval: 0.9cm
*Volume of the upper slice:
p . r² .height =
p X 5.2375 X 5.2375 X 0.9
cm³ =
77.560cm³
*Calculations according to Simpson’s formula:
diameter 
radius 
surface 
xfactor 
Result 
(cm) 
(cm) 
(cm²) 

(cm²) 





10.400 
5.200 
84.949 
1 
84.949 
10.245 
5.123 
82.435 
4 
329.742 
10.000 
5.000 
78.540 
2 
157.080 
9.500 
4.750 
70.882 
4 
283.529 
9.050 
4.525 
64.326 
2 
128.652 
8.300 
4.150 
54.106 
4 
216.424 
7.460 
3.730 
43.709 
2 
87.417 
6.140 
3.070 
29.609 
4 
118.436 
2.600 
1.300 
5.309 
1 
5.309 

1/3 X 0.9cm X 1411.538cm²
= 423.461cm³
TOTAL
VOLUME =
501.022cm³
2. Calculation
of the regression equation and the max content of the glass with the TI83
Conclusion : fairly strong correlation coefficient for
the quadratic regression but the quartic
regression model gives a better approximation
leads to a
max volume of 501.46277 cm³
3. Up to what height
must the glass be filled to obtain half of a full glass content ?
= ½ of the volume of a fully filled glass = 250.73139
a) With table (first AUTO to obtain a whole approximation,
afterwards with ASK)
b) and/or with SOLVE out of CATALOG
The glass should be filled up to 5.1 cm ( 5.0903 cm).
4. How many glasses can one get out of a magnum bottle (1.5 l) when the glass is filled
up to 90% ?
90% of the max volume = 0. 4513164929
l
Conclusion : out of a a magnum bottle (1.5 l) one can get ample
three (1500/451) ‘Westmalle’
glasses filled up to 90%.
7. CONTROL
*We carried out an experimental_control :
we filled a measuring glass with half a liter of water
and this volume filled up perfectly our ‘Westmalle’ glass.
*We also performed a calculated control making the sum of the volumes of all slides according
to the rule of Bezout.
r² . p . height = volume
VALUE OF THE VOLUME
(we make use of the mean)

IN
CM³ 
77.561 

74.991 

72.107 

67.196 

60.807 

53.195 

43.892 

32.685 

13.499 

SOM= 
495.933 
There is a small difference of 5cm³ between
the results of both calculations.
Conclusion : Simpson’s formula is
more accurate because the curvature of the glass is taken
in
consideration.
8. HISTORY
*Origin of the brewery Westmalle :
On April 22nd, 1836 the abbot de Ranc issued a decree stating that the monks had
to use the
popular drink from the region : water, skimmed milk, buttermilk and beer. The
works at the
brewery started on August 1st, 1836 and the first beer was ready on December 10th. Only round
about 1860 the beer was sporadicly sold to acquaintances and since 1872 it was
officially sold
to private persons. The real commercialisation started in 1920.
*On September 2nd, 1933 one voted for the construction of a new
brewery and on February
5th 1934 one started digging the foundations. Since 1992 the brewery is completely computer
controlled.
*Address : 496, Antwerpsesteenweg 2390 Westmalle (Belgium)
*Architect : Thomas Bas
*Maltmaster, construction engineer : Jan Adriaensens
*Staff : 36
*Production :
 3 100 000 qli of malt, 120 000 hl of beer,, water from the wells (70 m deep), malt of Dutch,
French and German origin, hop from Belgium, Germany, former Czechoslovakia, former
Yugoslavia, hop flowers, root sugar as a base product for the sugarcandy.
 The brewery is completely manned by laymen, except one : Father Lode,
commercial director
 Daily three brews of 215 hl in 12 hours
 Two mash coppers with direct flame on the copper tuns.
After filtration fresh yeast and sugar for fermentation in the bottle in
storehouses, 20 to 25°C,
two weeks for the for the tripel.
 98 % is sold in a bottle, 10 % for export
