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These exercises use Newton’s Law of Cooling. A hot bowl of soup is served at a dinner party. It starts to cool according to Newton's Law of Cooling so that i...

Question

These exercises use Newton’s Law of Cooling. A hot bowl of soup is served at a dinner party. It starts to cool according to Newton's Law of Cooling so that its temperature at time $t$ is given by $$T(t)=65+145 e^{-0.05 t}$$ where $t$ is measured in minutes and $T$ is measured in $^{\circ} \mathrm{F}$. (a) What is the initial temperature of the soup? (b) What is the temperature after 10 min? (c) After how long will the temperature be $100^{\circ} \mathrm{F} ?$

These exercises use Newton’s Law of Cooling. A hot bowl of soup is served at a dinner party. It starts to cool according to Newton's Law of Cooling so that its temperature at time $t$ is given by $$T(t)=65+145 e^{-0.05 t}$$ where $t$ is measured in minutes and $T$ is measured in $^{\circ} \mathrm{F}$. (a) What is the initial temperature of the soup? (b) What is the temperature after 10 min? (c) After how long will the temperature be $100^{\circ} \mathrm{F} ?$



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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