Differential Equations

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Newton's Law of Cooling

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A glass of hot water obeys Newton's law of cooling which postulates that the temperature of an object changes at a rate that is proportional to the difference between it and its surroundings:

$$\begin{equation} \frac{dT}{dt} = -k(T - T_0) \end{equation}$$

...where $T(t)$ is the temperature of the glass of water measured in Fahrenheit, $T_0$ is the (constant) temperature of the surroundings, $t$ is measured in minutes and $k$ is the proportionality constant.

Assume the glass is in a room that is at 70$^o$ F.

If initially the glass of water is 200$^o$F and one minute later, it is at 190$^o$F, the temperature of the water at any time $t$ (measured in degrees Fahrenheit) is:

A

$T(t) = 130 + 70 e^{-kt}$ and $k = \ln(13/12)$ per minute

B

$T(t) = 130 + 70 e^{-kt}$ and $k = \ln(12/13)$ per minute

C

$T(t) = 70 + 130 e^{-kt}$ and $k = \ln(13/12)$ per minute

D

$T(t) = 70 + 130 e^{-kt}$ and $k = \ln(12/13)$ per minute

E

There is not enough information in the question to determine the temperature