General Mathematics - Statistical Models - Rates of Cooling - Statistics Assignment Help

Assignment Task :

Description

The following data represents the temperature above room temperature of a cup of flat white coffee as it cools.

Where: A is the number of degrees above room temperature, and

t is the time in minutes that have passed since the coffee was poured.

In this task you will develop formulae to model the relationship between the number of degrees above room temperature of a cup of coffee and the time in minutes that have passed since the coffee was poured.

Section 1

1. Examine the data in the table above and predict, using only your ‘intuition’, what the temperature difference will be 2.5 minutes, 13 minutes, 35 minutes and 60 minutes after the coffee is poured. Predict how long it might take for the coffee to reach 5°C above room temperature.

2. On a large set of axes accurately plot the graph of A against t. Describe the shape of the graph and any of its significant features. Discuss whether it make sense to connect the dots with a continuous curve. 

3. Use your graph to estimate the temperature of the coffee above room temperature at the four different times given in part 1 above. Compare your predictions from part 1 with these answers.

4. Enter the tabled data into your graphics calculator and fit an appropriate model to it. Give the algebraic formula for this model. Discuss your reasons for choosing the model you used with reference to the features and behaviour of its graph. 

5. Use your model to predict the temperature of the coffee above room temperature for the same four times that were used in parts 1 and 3, as well as the time it would take for the coffee to reach 5°C above room temperature

6. Comment on similarities/differences in the temperature of the coffee as predicted using the three methods in parts 1, 3 and 5. Which values would you consider the most and least accurate predictions? Give clear reasons for your answers. Compare the two predictions made for the time it would take for the coffee to cool to 5°C above room temperature and comment on any difference between these two estimates.

Section 2

7. Consider what other factors might affect the rate of cooling of the coffee (e.g. size of the cup, what the cup is made of, initial temperature of the empty cup before the coffee is poured, ambient temperature of the room, type of coffee etc.). Choose one factor to investigate and describe in detail how you expect it might affect the graph of cooling of the coffee over time. 

8. Design an experiment where two variations of your chosen factor are used and all other parameters are kept consistent (e.g. you could make a cappuccino coffee and then a long black coffee keeping everything else about the experiment the same). Through direct measurement produce a table similar to the table above for each of these two new situations. 

9. Analyse your data using appropriate graphs and algebraic models. Compare any similarities and differences between the three models investigated. Conclude your investigation by summarising the important points discovered. Include discussion of the assumptions made in the investigation and the reasonableness of the models used.

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