Answer to Question #124063 in Differential Equations for Asubonteng Isaac Adjei

Question

A bowl of water has a temperature of 50◦C. It is put into a refrigerator where the temperature is 5◦C. After 0.5 hours, the water is stirred and its temperature is measured to be 20◦C. It is then left to cool further, take t in Newton’s law of cooling to be measured in minutes. Use Newton’s law of cooling to predict when the temperature will be 10◦C.

Answer 1

The Newton's law of cooling can be written in form

$T(t) = T_{\text{out}} + e^{-kt}(T_0-T_{\text{out}}).$ Here temperature is in Kelvins.

$293 = 278 + e^{-k\cdot 30}(323-278), \;\; e^{-30k} = \dfrac{15}{45}, \;\; k = 0.0366.$

$283 = 278 + e^{-kt}(323-278), \;\; e^{-kt} = \dfrac{5}{45}, \;\; t =60\,\mathrm{m} = 1\,\mathrm{h}.$