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Newton's Law of Cooling A thermometer reading $72^{\circ} \mathrm{F}$ is placed in a refrigerator where the temperature is a constant $38^{\circ} \mathrm{F}$ (...

Question

Newton's Law of Cooling A thermometer reading $72^{\circ} \mathrm{F}$ is placed in a refrigerator where the temperature is a constant $38^{\circ} \mathrm{F}$ (a) If the thermometer reads $60^{\circ} \mathrm{F}$ after 2 minutes, what will it read after 7 minutes? (b) How long will it take before the thermometer reads $39^{\circ} \mathrm{F} ?$ (c) Determine the time that must elapse before the thermometer reads $45^{\circ} \mathrm{F}$. (d) What do you notice about the temperature as time pass

Newton's Law of Cooling A thermometer reading $72^{\circ} \mathrm{F}$ is placed in a refrigerator where the temperature is a constant $38^{\circ} \mathrm{F}$ (a) If the thermometer reads $60^{\circ} \mathrm{F}$ after 2 minutes, what will it read after 7 minutes? (b) How long will it take before the thermometer reads $39^{\circ} \mathrm{F} ?$ (c) Determine the time that must elapse before the thermometer reads $45^{\circ} \mathrm{F}$. (d) What do you notice about the temperature as time passes?



Answers

Newton's Law of Cooling. According to Newton's law of cooling, if an object at temperature $ T $ is immersed in a medium having the constant temperature $ M $, then the rate of change of $ T $ is proportional to the difference of temperature $ M-T $. This gives the differential equation
$$ d T / d t=k(M-T) $$
(a) Solve the differential equation for $ T $
(b) A thermometer reading $ 100^{\circ} \mathrm{F} $ is placed in a medium having a constant temperature of $ 70^{\circ} \mathrm{F} $. After 6 min, the thermometer reads $ 80^{\circ} \mathrm{F} $. What is the reading after 20 min? (Further applications of Newton's law of cooling appear in Section 3.3.)


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