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.

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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.
Transcribed Image Text: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.

dTdt(T-T0)

OR

dTdt=k(T-T0)                 ..........(1)

Where,

T=Temperature of the bodyT0=Temperature of the surroundings=700 Ft=Timek=cooling constant

 

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

Integrating equation (1),we get

dT(T-T0)=kdt

Hence,

ln(T-T0)=kt+ln(C)         ........(2)

where,'C' is constant of integration.

Step 3:Apply Boundary Conditions and find unknowns.

First Boundary Conditions is,

t=0,T=1800F

Hence, from equation (2)

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

Hence,

C=1100 F       ...........(3)

 

 

Second boundary condition is,

t=10 min,T=1600F

hence, from equation (2) and (3)

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

Solving for 'k',

k=-0.02 min-1       ......(4)

 

 

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