Newton's Law of Heating/Cooling

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January 30th 2011, 03:02 PM
VonNemo19

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?
January 30th 2011, 03:05 PM
dwsmith

Quote:

Originally Posted by VonNemo19

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.
January 30th 2011, 03:11 PM
VonNemo19

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?
January 30th 2011, 03:25 PM
dwsmith

Quote:

Originally Posted by VonNemo19

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?