Céline Mortier and Karen Sijmons,  6 ECMT
Catharina Stappers, Abigail Van den Eynde, Dorien Van Poucke, 6 MW
                              

 

 

CONTENTS

  
1. 7-step 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.  7-STEP 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.yn-2 + 4.yn-1 + yn)
    whereby L is the interval

    Condition for applying the formula of Simpson is :there must be an odd number of 
    y-co-ordinates. 


    *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   cm
³   =    77.560cm³

   *Calculations according to Simpson’s formula:

 

diameter radius surface x-factor 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 TI-83

  

 

    
    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.                  

     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

  *Malt-master, 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 sugar-candy.
   - 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

 

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