Van Dosselaer Elsje  
6EcMt nr 18  
May 30rd, 2000
     

 

 

Bier met liefde gebrouwen,
drink je met verstand

       Beer brewed with love,
                             should be drunk with common sense
  

1. Problem

Choose a Belgian beer glass and determine the content using measuring points. With the help of a suitable regression function, to be found with the TI-83 calculator  and  the right integration calculations, this should lead to the maximum inner volume of the glass.

In addition the following questions should be solved :
1) Up to what height must the glass be filled to obtain half of a full glass ?
2) How many glasses can one get out of a Magnum bottle (1,5 l ) when the glass is filled up to 90 % ?

Because of the special shape of the glass and the fact that it is one of our most popular beers, I chose the ‘Leffe’ glass.

Leffe is thé market leader among the Belgian abbey beers.

To the best served chilled (5°C/41°F) with a large creamy head
 in the stylish Leffe chalice-type glass

Leffe Blond has a sunny golden yellow color.
 Its taste is light, aromatic, full-bodied delicate sweetness & flavor balance.
Leffe Blond has an alc./vol. of 6.6%.
 
Leffe Brown has a dark autumn brown color
 and a full-bodied taste, slightly fruity, with a hint of apple.
Leffe Brown has an alc./vol. of 6.5%.

Some other members of the Leffe-family are :
 Leffe Triple,  gold colouredr
with lemon and vanilla dominant in the taste,
supported by fresh, pure and light hints. It has an alc./vol. of 8.4%.
 Leffe Radieuse, which has  a yellow - brown color and a strong taste
with strong fruity hints and a rich bouquet.Leffe Radieuse has an alc./vol. of 8.2%.


2. Procedure

At first thought, I wanted to make several measurements at the outside of the glass with a flexible meter. But then I had to find a solution in order to deduct the thickness of the glass which has a rather irregular shape.

 
Finally I chose for the following method : 

1)      to start, I filled the glass with plaster

2)      when the plaster became solid, I removed(broke) the glass.

3)      afterwards, the middle of the bottom and top and their connecting line were marked

4)      finally, the circumference of several concentrical circles (always one cm higher starting from the bottom of the plaster cast ) were measured with the flexible meter. 

  

3. Measuring results 

Out of the formula of the circumference of a circle (2
pr), the inner radius of the glass(at severall heights) could be calculated.


  Circumference            =  2pr

        Height (cm)

   circumference/2p
                 = r

0,0

0,0

0,00

8,7

0,3

1,30

16,6

1,0

2,75

22,6

2,0

3,55

26,4

3,0

4,15

29,8

4,0

4,65

32,0

5,0

5,00

33,3

6,0

5,10

32,0

7,0

5,00

30,0

8,0

4,70

27,3

8,7

4,35

 

 

4. Calculations made with the graphical calculator TI-83

   a. Determination of the regression function :
      
*
with quartic regression

     

        with Ln regression  (starting from h = 0.1 with r = 0.1)    

  less correlation !

                 

    b. Calculations to find the max inner volume of the glass 
        and the solution of other related problems

       *  total inner volume of the glass

 
and   I = p*fn Int ( Y1 ,X, 0, 8.7)

 
gives 508.0325031 (in  cm )
starting from the quartic regression
 

            *  up to what height the glass must be filled to obtain half of the maximum volume ?

Starting from   Y2 = p*fn Int ( Y1 , X, 0,  X)

  we find with  SOLVE(Y2 – 254.0162516,X, 4, {3,8}) that  the glass must  be filled up  

to 5,368351905 cm height to obtain half of the maximum volume

            *  how many glasses, filled up to 90%, can one get out of a Magnum bottle (1,5 l) ?

90 % of the max inner volume is  508,0325031 cm x 0,90 = 457,2292528 cm

 Thus approximately 3,28 (1500 / 457,2292528) glasses, filled up to 90% of the inner volume

 of the Leffe glass, can one get out of the Magnum bottle.

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