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Math Help - Newton's Law of Heating/Cooling

  1. #1
    No one in Particular VonNemo19's Avatar
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    Newton's Law of Heating/Cooling

    Hi. Here's another problem:

    A thermometer reading 70 is placed in an oven preheated to a constant temp. Through a glass window in the oven door, an observer records that the oven reads 110 after .5 minutes, and 145 after 1 minute. How hot is the oven?

    So I begin by listing the given information:

    (1) T(0)=70
    (2) T(.5)=110
    (3) T(1)=145

    A one parameter family of solutions to Newton's Law is T(t)=ce^{kt}+T_m where T_m is the constant temperature of the surroundings - in this case, the oven.

    So, it seems like I would proceed by setting up and solving a system of three equations in three unknowns. Is this the correct method?
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  2. #2
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    Quote Originally Posted by VonNemo19 View Post
    Hi. Here's another problem:

    A thermometer reading 70 is placed in an oven preheated to a constant temp. Through a glass window in the oven door, an observer records that the oven reads 110 after .5 minutes, and 145 after 1 minute. How hot is the oven?

    So I begin by listing the given information:

    (1) T(0)=70
    (2) T(.5)=110
    (3) T(1)=145

    A one parameter family of solutions to Newton's Law is T(t)=ce^{kt}+T_m where T_m is the constant temperature of the surroundings - in this case, the oven.

    So, it seems like I would proceed by setting up and solving a system of three equations in three unknowns. Is this the correct method?
    I would just do this like your last problem and then take the limit as t goes to infinity to find the oven temp.
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  3. #3
    No one in Particular VonNemo19's Avatar
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    I'm not sure that I know how to do that. In the last problem, the temperature of the boiling water was known. How do I start?
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  4. #4
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    Quote Originally Posted by VonNemo19 View Post
    I'm not sure that I know how to do that. In the last problem, the temperature of the boiling water was known. How do I start?
    T(0)=c+T_m=70\Rightarrow c=70-T_m\Rightarrow T(t)=(70-T_m)e^{kt}+T_m

    \displaystyle T\left(\frac{1}{2}\right)=(70-T_m)e^{\frac{k}{2}}+T_m=110

    \displaystyle T(1)=(70-T_m)e^{k}+T_m=145

    \displaystyle e^{\frac{k}{2}}=\frac{110-T_m}{70-T_m}

    \displaystyle e^{k}=\frac{145-T_m}{70-T_m}\Rightarrow \left(e^{\frac{k}{2}}\right)^2=\frac{145-T_m}{70-T_m}

    Does this help?
    Last edited by dwsmith; January 30th 2011 at 04:30 PM. Reason: added extra hint.
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