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IB Mathematical Studies SL 2008

Internal Assessment

Glenunga International High School

Investigation:

The rate of consumption of water from different beverage glasses.

Date submitted: September 2, 2008

Session: November 2008

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CONTENTS Cover page……………………………………………………………. 1 Contents page…………………………………………………………. 2 Statement of task / Plan……………………………………………….. 3 Results and Mathematical Processes…………………………………. 5

Interpretation of results ………………………………………………18 Validity Discussion…………………………………………………...20 Bibliography…………………………………………………………..22

Appendix……………………………………………………………...23

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The rate of consumption of water from different beverage glasses.

Background Upon research, it was found that the shapes of beverage glasses are designed to limit the warming of wine, emphasise aromas and flavours, and preserve carbonation of champagne.1 It is also thought that the various shapes of beverage glasses direct the liquid within to a particular part of the mouth.2 An experiment was carried out at the University of Illinois which showed that the shape of beverage glasses and people’s perception influences how much they pour and drink.3 This investigation was inspired by the information described above and its general focus is the rate of consumption of water from different beverage glasses. Statement of task The first objective of this investigation is to determine whether a relationship exists between the different shapes of beverage glasses and the rate at which the beverage within is consumed. The second objective is to determine whether the rate of consumption decreases during drinking, i.e. after each swallow. Plan The investigation will be conducted using the following method:

1. Thirty different beverage glasses are selected. The pictures of these glasses can be found in the Appendix on pages 23 - 25.

2. The surface area of the top of each of the glasses is calculated. 3. A certain volume of water is added to each glass using a measuring device. The

volume does not need to be the same in each of the different glasses, e.g. 75ml in wine glass, 25ml in shot glass.

4. A drink of water is taken from each glass three times, without refilling the glass to the initial volume. N.B. For control purposes, only one individual will be taking a drink from each glass, as this keeps variables such as face dimensions and mouth volume constant.

5. The remaining content of each of the glasses is transferred back into the initial measuring device after each swallow. The volume is recorded and the difference between the initial volume and final volume is determined.

6. A chi-squared test is conducted in order to determine whether the rate of consumption for each glass is independent of the surface area of the top.

7. The results for the first swallow of each glass (surface area versus uptake of liquid) are plotted on a graph.

1 <URL: http://en.wikipedia.org/wiki/Wine_glass> 2 <URL: http://www.bestwineglass.com/pages/shapes___taste/29.php> 3 <URL: http://www.sciencedaily.com/releases/2003/10/031023070629.htm>

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8. The strength of the relationship is determined using Pearson’s Correlation Coefficient.

9. The appropriate line of regression is determined. 10. Differential calculus is used to determine the rate of consumption (volume per

cm2). 11. The results for the second and third swallows are graphed separately. The lines of

regression are determined and differentiated to find the rate of consumption. 12. The rates for each of the swallows are compared by substituting for x to determine

whether the rate of consumption decreases with drinking.

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RESULTS AND MATHEMATICAL PROCESSES SURFACE AREA OF TOP OF GLASS Example: Shot glass: Surface area of circle = πr 2

Diameter of top = 3.7cm π × (3.7÷2)2 = 10.75cm2

* The tabulated data below indicates that the initial volume of water in the shot glass is 25ml. After the first and second swallows, there are 14ml and 4ml remaining in the glass respectively. After the third swallow, there is no water left in the glass. The glass is not refilled during the drinking process.

RAW DATA

Table 1 – Amount of liquid remaining after each swallow (ml)

Type of glass /

surface area of top Initial amount of

liquid in glass (ml) 1st swallow

+1ml 2nd swallow

+1ml 3rd swallow

+1ml Shot glass / 10.75cm2

25 14 4 0

Red wine glass / 56.75cm2

75 46 21 0

Champagne glass / 30.19cm2

75 55 37 25

Liqueur glass / 31.17cm2

50 32 16 3

Beer glass / 40.72cm2

250 229 209 190

Rose coffee mug / 38.48cm2

250

228 207 190

Tia Maria glass / 38.48cm2

220

197 178 162

Crystal drinking glass / 32.17cm2

227

206 189 175

Espresso mug / 29.22cm2

60

43 27 17

Pink drinking glass / 45.36cm2

180

157 137 119

Silver goblet / 18.86cm2

35

22 11 6

Flinstones glass / 36.32cm2

170

148 132 117

Wide beer glass / 44.18cm2

325

301 281 263

*

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Christmas tea mug / 55.42cm2

250

222 198 178

Cappuccino cup / 44.19cm2

171

148 129 112

Orange juice glass / 40.72cm2

250

229 210 195

Nescafe mug / 38.48cm2

200

178 158 143

Mug (squares) / 50.27cm2

250

226 204 185

Crystal tumbler / 28.27cm2

350

332 316 305

Small tumbler / 22.90cm2

160

144 128 118

Doncafe small mug / 41.85cm2

170

150 131 116

Dog mug / 36.32cm2

250

230 212 199

Coffee mug white / 50.27cm2

240

215 192 172

Brown tea mug / 45.36cm2

200

177 157 141

Iced coffee glass / 44.18cm2

250

229 210 195

Glass mug / 35.26cm2

200

178 160 146

Simpson mug / 40.72cm2

250

225 204 187

Turkish coffee cup / 19.63cm2

36

22 10 3

Earl grey tea mug / 44.18cm2

210

188 166 148

Black coffee small cup / 43.01cm2

65

44 24 11

ERRORS AND UNCERTAINTIES The margin for error is + 1ml which was selected due to the whole number nature of the intake volume measurements. It is the smallest whole number value on the measuring device and therefore, to account for any errors, 1ml discrepancy is allowed each way. Amount of water consumed per swallow Example: Red wine glass: Water consumed in 1st swallow = 75ml – 46ml = 29ml

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PROCESSED DATA

Table 1 – Amount of liquid consumed per swallow (ml)

Type of glass / surface area of top

Initial amount of liquid in glass (ml)

1st swallow +1ml

2nd swallow +1ml

3rd swallow +1ml

Shot glass / 10.75cm2

25 11 10 4

Red wine glass / 56.75cm2

75 29 25 21

Champagne glass / 30.19cm2

75 20 18 12

Liqueur glass / 31.17cm2

50 18 16 13

Beer glass / 40.72cm2

250 21 20 19

Rose coffee/tea mug / 38.48cm2

250

22 21 17

Tia Maria glass / 38.48cm2

220

23 19 16

Crystal drinking glass / 32.17cm2

227

21 17 14

Espresso mug / 29.22cm2

60

17 16 10

Pink drinking glass / 45.36cm2

180

23 20 18

Silver goblet / 18.86cm2

35

13 11 5

Flinstones glass / 36.32cm2

170

22 16 15

Wide beer glass / 44.18cm2

325

24 20 18

Christmas tea mug / 55.42cm2

250

28 24 20

Cappuccino cup / 44.19cm2

171

23 19 17

Orange juice glass / 40.72cm2

250

21 19 15

Nescafe mug / 38.48cm2

200

22 20 15

Mug (squares) / 50.27cm2

250

24 22 19

Crystal tumbler / 28.27cm2

350

18 16 11

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Small tumbler / 22.90cm2

160

16 16 10

Doncafe small mug / 41.85cm2

170

20 19 15

Dog mug / 36.32cm2

250

20 18 13

Coffee mug white / 50.27cm2

240

25 23 20

Brown tea mug / 45.36cm2

200

23 20 16

Iced coffee glass / 44.18cm2

250

21 19 15

Glass mug / 35.26cm2

200

22 18 14

Simpson mug / 40.72cm2

250

25 21 17

Turkish coffee cup / 19.63cm2

36

14 12 7

Earl grey tea mug / 44.18cm2

210

22 22 18

Black coffee small cup / 43.01cm2

65

21 20 13

CHI-SQUARED TESTING H0 states that volume of intake of liquid is independent of the surface area of the top of the beverage glass. H1 states that volume of intake of liquid is dependent on the surface area of the top of the beverage glass. Degrees of freedom = (r − 1)(c − 1) = (2 − 1)(2 − 1) = 1 × 1 df = 1 The test is conducted at 10% significance level, as this takes into account the uncertainty of the relationship being present in a majority of the sample, due to a number of influences that could affect the trend obtained. As only one trial was conducted to control variables, the 10% significance level reflects the likelihood of surface area having an impact on volume of intake of liquid, i.e. it shows the relationship for 90% of a large sample. We reject the null hypothesis H0 if χ2

calc is greater than 2.706, which is the critical value for the 10% significance level with 1 degree of freedom.

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Contingency Table (based on first swallow) < 40cm2 > 40cm2 Sum

< 22ml 10 5 15 > or equal to 22ml 5 10 15

Sum 15 15 30 These intervals are used because they cause the frequencies to be greater than or equal to 5. Considering the size of the sample and the fact that the chi-squared test is for large samples, the table should not have frequencies of less than 5, as these values are considered too small and would not accurately reflect the relationship between the two variables. Expected Frequency Table < 40cm2 > 40cm2 Sum

< 22ml 15 × 15 = 7.5 30

15 × 15 = 7.5 30

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> or equal to 22ml 15× 15 = 7.5 30

15 × 15 = 7.5 30

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Sum 15 15 30 Chi-squared Table

fo fe fo − fe (fo − fe)2 (fo − fe)

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fe 10 7.5 2.5 6.25 0.833 5 7.5 -2.5 6.25 0.833 5 7.5 -2.5 6.25 0.833 10 7.5 2.5 6.25 0.833

Total 3.332 The χ2

calc for this set of data is 3.332. As χ2calc is greater than the 10% significance level

critical value, H0 is rejected. Thus, the volume of intake of liquid is dependent on the surface area of the top of the glass for 90% of a large sample.

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PEARSON’S CORRELATION COEFFICIENT The Pearson’s Correlation Coefficient for the set of data representing the first swallow is calculated to determine whether the relationship between the surface area of the top of the glass and the volume of intake is linear. The scatterplot below shows the relationship between surface area and volume of intake. Pearson’s Correlation Coefficient will be determined from the 30 data points. The method and the result obtained follows.

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x y xy x2 y2

10.75 11 118.25 115.56 121 56.75 29 1645.75 3220.56 841 30.19 20 603.8 911.44 400 31.17 18 561.06 971.57 324 40.72 21 855.12 1658.12 441 38.48 22 846.56 1480.71 484 38.48 23 885.04 1480.71 529 32.17 21 675.57 1034.91 441 29.22 17 496.74 853.81 289 45.36 23 1043.28 2057.53 529 18.86 13 245.18 355.70 169 36.32 22 799.04 1319.14 484 44.18 24 1060.32 1951.87 576 55.42 28 1551.76 3071.38 784 44.19 23 1016.37 1952.76 529 40.72 21 855.12 1658.12 441 38.48 22 846.56 1480.71 484 50.27 24 1206.48 2527.07 576 28.27 18 508.86 799.19 324 22.9 16 366.4 524.41 256 41.85 20 837 1751.42 400 36.32 20 726.4 1319.14 400 50.27 25 1256.75 2527.07 625 45.36 23 1043.28 2057.53 529 44.18 21 927.78 1951.87 441 35.26 22 775.72 1243.27 484 40.72 25 1018 1658.12 625 19.63 14 274.82 385.34 196 44.18 22 971.96 1951.87 484 43.01 21 903.21 1849.86 441

1133.68 629 24922.18 46120.76 13115

y = Σy n

= 629 30

= 20.97

x = Σx n = 1133.68 30 = 37.79

Σ

First, the mean volume of intake is calculated:

Now, the mean surface area is calculated:

The results for Pearson’s Correlation Coefficient are shown on the following page.

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The scatterplot for the data is shown on the next page. The mean point is calculated by finding the mean of the logS values and the mean of the logV values using the formulae ΣlogS and ΣlogV respectively. n n Mean of logS = 46.6603 30 Mean of logV = 39.3856 30

r = Σxy − nxy . √Σx2 − nx2 × √Σy2 − ny2 = 24922.18 − (30 × 37.79 × 20.97) . (√46120.76 − 30 × 37.792) × (√13115 − 30 × 20.972) = 0.9395 r2 = (0.9395)2

= 0.883 The r2 value indicates the strength of association between the dependent variable and the independent variable. If there is a causal relationship then r2 indicates the degree to which change in the independent variable explains change in the dependent variable. For example, r2 = 0.883 for a linear relationship implies that 88.3% of the variation in volume of intake can be explained by the variation in the surface area of the top of beverage glasses. This value confirms that there is a strong correlation between the surface area of the top of glasses and the volume of intake of liquid. However, while this value is relatively strong enough to indicate a linear relationship between the two variables, it is assumed that the relationship would not be linear, as the volume of intake must plateau at a certain point due to the concept of drinking comfort and mouth volume. The incidence of these additional variables is a cause for further investigation into the relationship between surface area and volume of intake. The r2 value for other possible relationships will also be determined using a Graphics Display Calculator (GDC) in order to ascertain whether there is a stronger relationship between the variables. The strongest r2 value is 0.909 which represents a power relationship and indicates that the trend of the scatterplot has slight curvature, rather than perfect linearity. The equation of this relationship will be determined by plotting logS (the logarithms in base 10 of the surface area values) against logV (the logarithms in base 10 of the volume of intake values) and using the general equation y = mx+c to find the equation of the line of best fit. This will be done by finding the slope between two points that lie on the line of best fit (which is generated by finding the mean point and predicting a regression line). From this gradient, the y-intercept (c) will be determined, and thus, an equation for the graph will be determined, using logarithm laws.

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The mean of the logS values is 1.56 and the mean of the logV values is 1.31. The regression line is predicted by drawing a line passing through the mean point and as many data points as possible.

Relationship between logS and logV

The graph above shows a strong positive correlation between the logS and logV values, indicating that the variables are related.

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Using the graph, the regression line passes through the points (1.2929, 1.1461) and (1.7013, 1.3979). Therefore, using the general equation y = mx+c : m = 1.3979 − 1.1461 1.7013 − 1.2929 = 0.617 y = 0.617x + c Now, substitute the point (1.2929, 1.1461) to find c. 1.1461 = 0.617(1.2929) + c 1.1461 − 0.7977 = c c = 0.348 Thus, the equation is y = 0.617x + 0.348. Now, logarithm laws will be used to find the equation of the power relationship between surface area (S) and volume of intake (V). All logs are in base 10. logV = 0.617logS + 0.348 {logAn = n} = logS0.617 + log100.348 {nlogA = logAn} = log (S0.617 × 100.348) {logA + logB = log(A×B)} V = S0.617 × 100.348 {alog

ax = x}

= S0.617 × 2.23 Therefore, the equation of the power relationship is V = 2.23×S0.617. Now, to compare the equation calculated manually with results from technology, the equation above will be graphed alongside the equation generated by the Graphics Display Calculator (V = 2.66×S0.571), on FX Draw.

V = 2.23×S0.617

V = 2.66×S0.571

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The graph shows that as the values get larger, the differences between the manually generated equation and the equation generated using technology increase. To maintain accuracy, the equation determined by using technology will be used, and technology will also be used to calculate the equations of the second and third swallows. DIFFERENTIAL CALCULUS Now, to find the rate of consumption of liquid from the glasses, the line of regression for the power relationship between surface area of the top of the glass and volume of intake will be differentiated. Let f (S) = 2.66×S0.571 f '(S) = 1.52×S−1.571

Let S = 1 f '(S) = 0.518ml per cm2

The table alongside shows that the rate of consumption decreases as surface area gets larger. This may be due to the increased difficulty in drinking from a glass which has a large surface area on top, as liquid cannot be directed to the mouth as easily. The graph below illustrates the decreasing rate of consumption as surface area increases. It appears that the graph eventually plateaus, which could suggest the involvement of a number of factors that will be

discussed in the interpretation section.

Surface area (cm2) Slope 20 0.0137 25 0.00968 30 0.00727 35 0.00570 40 0.00462 45 0.00384 50 0.00326

This value shows that when surface area (S) = 1, the liquid is being consumed at the rate of 0.518ml/cm2 in the first swallow.

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Therefore, in conclusion, the first objective of this investigation has been satisfied. The surface area of the top of beverage glasses has an effect on the rate of consumption of the liquid within. As surface area increases, the rate of consumption decreases, possibly due to the concept of drinking comfort, i.e. as the surface area of the top of the glass gets larger, it becomes more difficult to drink and thus the volume of intake per cm2 decreases. In order to determine whether the rate of consumption of water decreases after a number of swallows, the relationship between surface area and volume of intake for each swallow will be graphed. The process for determining the line of regression and r2 value is as shown above, but the Microsoft Excel trendline tool will be used for the graphs of the second and third swallows, as this gives a more accurate result.

Line of regression = 2.4066x0.565 Using differential calculus, the rate of consumption during the second swallow is f '(x) = (0.565 × 2.4066)x0.565 – 1 = 1.36x-1.565

Surface area (cm2) Slope

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As for the first swallow, the data for the second swallow indicates that the rate of consumption decreases with an increase in surface area. This is shown by a decrease in the slope of the regression line as surface area increases. This relationship supports the idea that drinking comfort may be a determinant of the rate of consumption from different beverage glasses.

Line of regression = 0.3036x1.0651 Rate of consumption = (0.3036 × 1.0651)x1.0651 - 1 = 0.323x0.651

The slopes for swallow 3 increase with surface area. As this is not consistent with previous swallows, some factor is responsible for an increase in the rate of consumption as surface area increases. A possible explanation is that the liquid remaining in smaller surface area glasses is minimal, and thus, only a certain amount can be consumed per cm2, whereas more can be consumed from glasses with greater surface area.

20 0.0125 25 0.00883 30 0.00664 35 0.00521 40 0.00423 45 0.00352 50 0.00298

Surface area (cm2) Slope 20 2.27 25 2.63 30 2.96 35 3.27 40 3.57 45 3.85 50 4.12

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INTERPRETATION OF RESULTS There is a very strong power relationship between surface area of the top of glasses and the volume of intake of liquid. The 10% significance test is used to rule out the probability of the relationship having occurred by chance, and supports the fact that there is a strong relationship between the variables. There is 90% certainty that the relationship between the variables is valid and 10% likelihood that the results occurred by chance. The chi-squared result indicates that as the surface area of the glass increases, the volume of intake increases. However, due to the fact that the test of independence was conducted at 10% significance level, there is the possibility of there being no relationship between the variables in 10% of a large sample. This could be explained by taking into consideration a number of variables that could affect the rate of consumption of water from different glasses, which will be discussed in the validity section of this investigation. The results for the second objective of the investigation show that the rate of consumption does not decrease during drinking. This could be due to fluctuations in the level of thirst, which would have influenced the volume of intake for the second and third swallows. Perception may also have been a factor responsible for the trend shown over the three swallows. As the shape of the glass was noted to have an effect on how much people pour and drink in the Illinois University study, it is also quite possible that perception has an effect on drinking habits.4 For example, if there appears to be a substantial volume of liquid left in the glass, then the individual may be likely to drink more during the second or third swallow or otherwise, if there is very little liquid remaining, then the individual may consume considerably less. The scatterplot showing surface area against volume of intake indicates that, in general, as surface area increases, so does the volume of intake. Eventually, the volume of intake reaches a plateau, suggesting that surface area no longer impacts how much is consumed. This could be due to the comfort factor that will be discussed below. The use of differential calculus to determine the rate of consumption (ml per cm2) shows that the rate decreases during the first and second swallows. This could possibly be explained by examining the concept of drinking comfort. Hypothetically speaking, if an individual is drinking from a salad bowl their ability to tilt their head back to consume the liquid without spilling any would be difficult, and thus their rate of consumption (ml per cm2 of surface area) would be slower than if they were drinking from a drinking glass. When the slope is graphed against surface area, there is an exponential decay relationship which suggests that the rate of consumption eventually plateaus. This means that at very large surface areas, the rate of consumption does not change significantly, as suggested by the ‘drinking comfort’ theory. The results for the second objective of this investigation; whether the rate of consumption decreases during drinking, indicate that the rate generally decreases. However, the third swallow shows an increasing slope which could be due to perception affecting how much

4 <URL: http://www.sciencedaily.com/releases/2003/10/031023070629.htm>

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people take in from glasses with larger surface area, i.e. they may appear to have more liquid within them. The volume of intake for each of the glasses may have been impacted by the personal drinking habits of the individual undertaking the experiment. For example, if the individual is primarily a coffee drinker, the volume of liquid taken in from a coffee mug could have been affected by the fact the individual is likely to consume a certain volume according to personal drinking habits. This also validates the use of the 10% significance level in the chi-squared test, as individual habits may cause variation in the rate of consumption.

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VALIDITY The reliability of the results obtained through this investigation may have been compromised by the lack of replication with other individuals. However, this method was applied in order to control variables that may have otherwise impacted the final results; for example, face dimensions and mouth volume, which differ for individuals and thus could have affected the results obtained. The chi-squared test of independence is inaccurate for 10% of a large sample, and this has an impact on the validity of the results obtained. If 10% of a sample does not indicate a relationship between surface area and volume of intake, then the validity of the conclusions drawn from this investigation are somewhat compromised. The chi-squared test is for large samples and as the sample for this investigation is relatively small, this also questions the reasonableness of the calculations and models generated. However, due to the large number of variables involved in the investigation which had to be kept constant, the use of the 10% significance level is justified. The inconsistent swallow habits of individuals also justifies the need for testing at a 10% significance level, as this allows for consideration of large differences in personal habits. The mathematical processes used ensure that the data has been interpreted and communicated in a relevant way. As the study focuses primarily on the surface area of the tops of different beverage glasses and the rate of consumption of liquid within, the use of trendlines and differential calculus is justified. Although the sample is relatively small, the results are reasonable because they indicate a relationship, which is suggested by the background information acquired. The chi-squared test does not necessarily undermine the reasonableness of the results because the investigation deals with a number of variables that could significantly impact results generated by replication. Overall, the investigation is limited in its inability to suggest with 100% certainty that a relationship exists between surface area and rate of consumption, but this is to be expected considering individual habits and face dimensions. The main limitation of this investigation is the use of water in the trials, as this is not necessarily indicative of people’s drinking habits. The type of liquid in each of the glasses is a significant factor affecting the rate of consumption. If, for example, the wine glass contains 75ml of expensive red wine, then there is the likelihood of the drinker subconsciously limiting intake in order to prolong the volume of liquid available to them. However, due to the controlled nature of the investigation, the use of different types of liquid would create difficulty in interpretation of results, particularly in terms of concluding whether the rate of consumption is related to surface area. Other factors that may also affect the rate of consumption, including viscosity of liquids and temperature of liquids, justify the need for controlled liquid in the samples, i.e. water. If one sample is hot coffee, this evidently changes the rate of consumption from the coffee mug. Similarly, if a milkshake is compared to water, the milkshake is more viscous and as such, the rate of consumption would be lower than for the water sample. Drinking in succession also contributes to the variable of ‘thirst’, that is, if an individual is drinking from one glass consecutively (due to time restraints, it was unrealistic to allow a large

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period of time to drink from one glass) then it is likely that their thirst levels decrease, thus limiting their intake and consequently affecting the rate of consumption. A further improvement that could be made to the investigation in order to maintain validity is to select beverage glasses with a greater variation in surface area, i.e. more glasses with a surface area greater than 50cm2 and less than 20cm2. A large variation in the surface area of the top of glasses would indicate whether the results obtained and conclusions drawn are realistic. Furthermore, a larger sample of different types of beverage glasses, i.e. more than 30 different types, would also validate the mathematical processes used (trendlines and chi-squared testing specifically) and ensure that the results obtained from the use of these processes are accurate and justified.

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BIBLIOGRAPHY Haese (2004), Mathematics for the international student: Mathematical Studies SL, Haese & Harris Publications, Adelaide, Australia. Haese (2007), Mathematics for year 12: Mathematical Applications – Second Edition, Haese & Harris Publications, Adelaide, Australia. “Shape of Beverage Glass Influences How Much People Pour and Drink” [online] [Accessed 8th May] Available from World Wide Web: <URL: http://www.sciencedaily.com/releases/2003/10/031023070629.htm> “Wine glass” [online] [Accessed 7th May] Available from World Wide Web: <URL: http://en.wikipedia.org/wiki/Wine_glass> “Wine Glass Shapes and Taste” [online] [Accessed 7th May] Available from World Wide Web: <URL: http://www.bestwineglass.com/pages/shapes___taste/29.php>