word problem- temp

Question

A roasted turkey is taken from an oven when its temperature has reached 185 Fahrenheit and is placed on a table in a room where the temperature is 75 Fahrenheit. Give answers accurate to at least 2 decimal places.

(a) If the temperature of the turkey is 156 Fahrenheit after half an hour, what is its temperature after 45 minutes?

(b) When will the turkey cool to 100 Fahrenheit?

Answer 1

Use Newton's Law of Cooling

 

Where temp. Is a function of time

 

f[T(t)] and T(t)=Ta+(To-Ta)e^-kt

 

Where t is time elapsed, a is ambient temp, o is when start time, the constant e, k is the differential constant.

 

Given.

 

To=185, Ta=75, Tt=30 min

Solve for f(45)

 

Fisrt solve for 30 min to get k constant

 

f(156), T(30)=T75+(T185-T75)e^-kt. Remember T(t) that (t)=30

 

156=75+(185-75)e^-kt

81=110e^-k(30). This is where you put the t that is 30

81/110=e^-30k. Invert both sides to get positive exponent

110/81=e^30k. Take ln of both sides to get rid of e.

ln(110/81)=ln(e^30k)

ln1.36=30k

k=ln1.36/30=(.01)

 

Now plug in k

 

f[T(45)]=75+110e^-(.01)(45)=75+ 20.03=95.03F at 45 min

 

Now solve for 100F,notice it must occur before 45 min and after 30 min.

 

f(100), T(t)=25+110e^-(.01)t use law of logorithms

 

25=110e^(-.01)t

25/110=e^-(.01)t

ln(110/25)=ln(e)*[t/100]

ln(ln110/ln25)=t/100

t=100*ln(ln110/ln25)

t=38.29

 

 

 

 Plug in new t for f(100)

 

T(38.29)=25+110e^-(.01)(38.29)= 100

 

Therefore

 

At T(45min)=95.03F

At 100F, t=38.29min