Final work : Mathematics

 

 

 

    

 

 

 

 

 

 

Speleman Eefje                                                                            6MWa

Van den Bogaert Sarah                                                              OLVP Sint-Niklaas

Van Soye Katlijn                                                                          June 2000

 

An application of the integrals and statistics with the graphical calculator TI-83

 

1.     Determination of the co-ordinates of several points located on the  Duvel beer glass

    To find the co-ordinates of several points on the Duvel glass, we followed two procedures:

  1. Having the glass filled with white sand, we placed it before a sheet of paper where the sunrays marked his shadow. Because we couldnít investigate properly whether the runrays fell in vertically on the glass, it seemed to be too difficult to make an accurate drawing of the shape of the glass 
  2. We thought better of measuring the circumference with a string at different heights of the glass and using the formula :

 

height          circumference         radius
0                              4.02                      0.64
1                            14.01                      2.23
2                            24.00                      3.82
3                            26.51                      4.22
4                            29.02                      4.62
5                            28.27                      4.50
6                            27.02                      4.30
7                            24.50                      3.90
8                            23.50                      3.74
9                            21.99                      3.50
10                          20.98                      3.34
11                          21.99                      3.50
12                          22.49                      3.58
13                          24.00                     
3.82

                                        

 

 The lists of  heights resp radius were entered in the calculator with  STAT ŗ Edit   (See 1a, 1b)  

  

2.     Regression

       Calculator   :    STAT ŗ Calculate   ŗ   7. Quart Reg   ŗ   L1, L2, Y1    (See 2a,2b)

       The Quartic Regression ( for graph See 3a) gives the best result
       With correlation:    r≤ = 0.9908935551   (See 2b)                                     
       Conclusion : very good correlationcoŽfficiŽnt      

 

3.     Function

        From the calculator we get a function of the 4th degree : f(x) = ax 4  + bx3 + cx2 + dx + e 

              
      

4.     Calculation of the maximum inner volume of the glass

          

    

      Calculator: 

                   

       Volume573.2812403 cm≥  =  0.5732812403 dm≥  =  0.5732812403 l    

       Conclusion: The Duvel glass filled to the maximum holds 0.573 liter  

  

5.     Up to what height must the glass be filled to obtain half of a full glass content?
 

   * For half the volume of the glass we use:                                                                                           

         

         

    

       Conclusion:   The Duvel glass is filled up to 50% containing 286.406202 ml of liquid                                                      

  * To find the corresponding height we solve :   

     

       and get  h = 6.1623 cm                        

       Calculator (See 4a, 4b, 4c en 4d):

        

                          X                     Y2

                     6.1622                          286.63
                     6.1623                          286.64
                     6.1624                          286.64
                     6.1625                          286.65

        Conclusion:  to fill the Duvel glass up to 50%,  the liquid must be up to a height of  6.16 cm  

 

 6.     How many glasses can one get out of a Magnum bottle (1.5 l) when the glass must 
       be filled up to 90%  ?   

        90% of the total inner volume is  :    573.2812403 * 90% = 515.9531163

        and        1500 / 515.9531163 = 2.907240896

        Conclusion: Out of the Magnum bottle we can fill almost 3 Duvel glasses filled up to 90% . 

 

7.     Experiment of control guided by chemistry teacher Mr  Benoit Naessens

  

 

 

  

      Note : Since we performed our measurments from the outside of the glass (we didnít  
      substract the thickness of the glass),  the volume obtained by our previous calculations is

      a little bit  larger than the volume of the liquid we could actually pour in the glass.

 

Print 1a en 1b : list L1 with the heights measured and L2 with the corresponding radius

Print 2a en 2b : screen prints of the  quartic regression

 

Print 3a en 3b : graph containing the measured data and the regression curve

Print 4a, 4b, 4c en 4d : entering Y2, we can read the asked height from the table

 

 

 

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