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Math Help - The Cooling Law Problem

  1. #1
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    Red face The Cooling Law Problem

    A temperature probe is placed in a saucepan filled with hot water, and the temperature is recorded at 30 second intervals for 5 minutes. The ambient room
    temperature is 18.5.
    The data is recorded in the table.
    Time (t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
    Temperature (oC) 98 91.5 87.5 84 80.5 78 75.5 73 71 69.5 68
    Note: For all following questions all working out must be shown as justification

    a) Use the standard exponential regression model (y=abx) to develop an equation in this form for Temperature as a function of Time (min). What temperature does this predict after 30 minutes?

    b) What is worrying about this prediction?
    This raises a worrying aspect of fitting regression models to data. An exponential relationship seems plausible, and yet the suggested one has limited predictive power. Indeed other regression models fit the data well, but are not helpful as predictors for the situation (see below).
    The Cooling Law Problem-image.jpg

    c) Develop a model by creating a new list of 'excess' temperatures (T - 18.5), and again doing the standard exponential regression on (T - 18.5) versus t. What temperature does this predict after 30 mins? How does this compare with the model from part a)
    d) From an analysis of your answers in part a), b) and c), justify when it would be best to add the milk to the cup of black coffee to give the best chance of a nice hot cup of coffee if the phone rings : before answering the phone or after the brief phone call has ended.
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  2. #2
    MHF Contributor

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    Re: The Cooling Law Problem

    Quote Originally Posted by nabey1 View Post
    A temperature probe is placed in a saucepan filled with hot water, and the temperature is recorded at 30 second intervals for 5 minutes. The ambient room
    temperature is 18.5.
    The data is recorded in the table.
    Time (t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
    Temperature (oC) 98 91.5 87.5 84 80.5 78 75.5 73 71 69.5 68
    Note: For all following questions all working out must be shown as justification

    a) Use the standard exponential regression model (y=abx) to develop an equation in this form for Temperature as a function of Time (min). What temperature does this predict after 30 minutes?
    Okay, you have to determine two values, a and b, so you need two equations. Typically, it is best two use two distant points on the graph, here, x= 0, y= 98 and x= 5, y= 68.
    That is, solve 98= ab^0 and [itex]68= ab^5[/itex] for a and b. Once you have found a and b, determine ab^{30}.

    b) What is worrying about this prediction?
    This raises a worrying aspect of fitting regression models to data. An exponential relationship seems plausible, and yet the suggested one has limited predictive power. Indeed other regression models fit the data well, but are not helpful as predictors for the situation (see below).
    Is this your answer? It's not clear to me what you are saying. Why does "the suggested one" have limited predictive power? Why are the other models you show not helpful?

    Click image for larger version. 

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    c) Develop a model by creating a new list of 'excess' temperatures (T - 18.5), and again doing the standard exponential regression on (T - 18.5) versus t. What temperature does this predict after 30 mins? How does this compare with the model from part a)
    d) From an analysis of your answers in part a), b) and c), justify when it would be best to add the milk to the cup of black coffee to give the best chance of a nice hot cup of coffee if the phone rings : before answering the phone or after the brief phone call has ended.
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