Mathematics

Mathematical Studies

Investigation Folio 1

Cooling curves

Cindy Chung Cia Yung

1301G22658

SP1

Miss Christine

In this folio, I will be investigating the cooling rates of various beverages and explain the significance of this investigation. This investigation will be on hot white coffee, in part A with data that has been provided and then on tea and Milo, in part B, with data that was collected by my group. A hot object specifically a hot beverage in this case, has a higher temperature than the thermal equilibrium. With the data, a mathematical model of best fit will be drawn and from that a conclusion to the cooling rate of that beverage will be formed.

The temperature is taken every 5 minutes over a 60 minute period. The room temperature at the time of data collection was also recorded. Different beverages are used due to the fact that each beverage has a specific recipe, the ingredients can affect the cooling rate of the beverages therefore to minimize these differences, and we can average the cooling rates of all three beverages and obtain an approximate value for the cooling rates of liquids.

Part A

The data for the cooling rate of a cup of hot white coffee is given as follows (TR was 28 OC):

Time, t (minutes) | Temperature, TB (OC) |

0 | 80.0 |

5 | 69.0 |

10 | 64.0 |

15 | 58.0 |

20 | 55.5 |

25 | 53.0 |

30 | 50.0 |

35 | 47.5 |

40 | 45.0 |

45 | 43.0 |

50 | 42.0 |

55 | 40.5 |

60 | 39.5 |

The scatterplot drawn from this set of data is shown below. (Figure 1.1)

(Figure 1.1)

Adam’s model is superimposed on the scatterplot below. Adam’s equation is

y = 0.0132x2 – 1.43x + 79.7. (Figure 1.2)

(Figure 1.2)

Adam’s model of best fit passes through 2 points of the scatterplot, this shows that it is still possible for the model to be the model of best fit but this model is a quadratic model (parabolic shape) which can be deduced by the equation given. A quadratic model cannot fit the scatterplot given because the...