When an object is removed from a furnace and placed in an environment with a constant temperature of 90 degrees Fahrenheit , its core temperature is 1120 degrees Fahrenheit . Find the core temperature 5 hours after the object is removed from the furnace.

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When an object is removed from a furnace and placed in an environment with a constant temperature of 90 degrees Fahrenheit , its core temperature is 1120 degrees Fahrenheit . Find the core temperature 5 hours after the object is removed from the furnace.

Mathematics
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This should be an application of Newton's Law of Cooling. The law says that the rate of change in the temperature of the object is proportional to the difference between the object temperature and the environment; that is,\[\frac{dT}{dt}=-k(T-T_{\infty})\]where k is positive, T is the object temperature and T_{infinity} that of the environment. This is a separable equation and you have\[\frac{dT}{T-T_{\infty}}=-kdt \rightarrow \log(T-T_{\infty})=-kt+c\]Exponentiating both sides gives,\[T-T_{\infty}=Ce^{-kt} \rightarrow T= Ce^{-kt}+T_{\infty}\]At t=0, the temperature of the object is its initial temperature, and so,\[T(0)=C+T_{\infty} \rightarrow C=T(0)-T_{\infty}\]Your equation is then\[T(t)=[T(0)-T_{\infty}]e^{-kt}+T_{\infty}\]Now, the problem you have here is that there isn't enough information to extract a numerical answer. We need some more info. to find k. From what you do have,\[T(t)=(1120-90)^oFe^{-kt}+90^oF\]\[=(1030^oF)e^{-kt}+90^oF\]At t=5 hours,\[T(5)=(1030^oF)e^{-5k}+90^oF\]

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