5. You are cooling a dish that was taken out of the oven with an internal temperature of 180°F.After 10 minutes outside in a 70°F apartment, the temperature of the dish has reached 160°F.(a) Using Newton's Law of Cooling, set up a differential equation and show why temperaturey of the dish after t minutes out of the oven, is described by the function y(t) = 70+110ektwhere k is the cooling constant (you must find).(b) If the dish is safe to eat at 120°F, use the function in (a) to predict how long it needs tobe left outside the oven.

Answered: 5. You are cooling a dish that was…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

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5. You are cooling a dish that was taken out of the oven with an internal temperature of 180°F.
After 10 minutes outside in a 70°F apartment, the temperature of the dish has reached 160°F.
(a) Using Newton's Law of Cooling, set up a differential equation and show why temperature
y of the dish after t minutes out of the oven, is described by the function y(t) = 70+110ekt
where k is the cooling constant (you must find).
(b) If the dish is safe to eat at 120°F, use the function in (a) to predict how long it needs to
be left outside the oven.

Expert Solution

Step 1:Write Newton's Law of cooling.

According to Newton's law of cooling, the rate of loss of a body is directly proportional to the difference of the temperatures between the body and its surroundings.

Hence,Mathematically.

$\frac{d T}{d t} \propto (T - T_{0})$

$\frac{d T}{d t} = k (T - T_{0})$ $. . . . . . . . . . (1)$

Where,

$T = T e m p e r a t u r e o f t h e b o d y T_{0} = T e m p e r a t u r e o f t h e s u r r o u n d i n g s = 70^{0} F t = T i m e k = c o o l i n g c o n s \tan t$

Step 2:Find temperature of the body (T) at any time 't'.

Integrating equation (1),we get

$\int \frac{d T}{(T - T_{0})} = \int k d t$

Hence,

$\ln (T - T_{0}) = k t + \ln (C)$ $. . . . . . . . (2)$

where,'C' is constant of integration.

Step 3:Apply Boundary Conditions and find unknowns.

First Boundary Conditions is,

$t = 0, T = 180^{0} F$

Hence, from equation (2)

$\ln (180 - 70) = k (0) + \ln (C)$

Hence,

$C = 110^{0} F$ $. . . . . . . . . . . (3)$

Second boundary condition is,

$t = 10 m i n, T = 160^{0} F$

hence, from equation (2) and (3)

$\ln (160 - 70) = k (10) + \ln (110)$

Solving for 'k',

$k = - 0.02 m i n^{- 1}$ $. . . . . . (4)$

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