SOLUTION: The temperature of a roast varies according to Newton's Law of Cooling: dT/dt= -k(T-A), where T is the water temperature, A is the room temperature, and k is a positive constant. I

Algebra ->  Rational-functions -> SOLUTION: The temperature of a roast varies according to Newton's Law of Cooling: dT/dt= -k(T-A), where T is the water temperature, A is the room temperature, and k is a positive constant. I      Log On


   

Question 1085510: The temperature of a roast varies according to Newton's Law of Cooling: dT/dt= -k(T-A), where T is the water temperature, A is the room temperature, and k is a positive constant. If a room temperature roast cools from 68°F to 25°F in 5 hours at freezer temperature of 20°F, how long (to the nearest hour) will it take the roast to cool to 21°F?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
dT%2Fdt=-k%28T-A%29
Let's do a substitution,
U=T-A
dU=dT
So then,
dU%2FU=-kdt
ln%28U%29=-kt%2BC
U=Ce%5E%28-kt%29
T-A=Ce%5E%28-kt%29
and
T%280%29=68
T%285%29=25
So,
68-20=Ce%5E%28-k%280%29%29
C=48
and
25-20=48e%5E%28-5k%29
5=48e%5E%28-5k%29
e%5E%28-5k%29=5%2F48
-5k=ln%285%2F48%29
k=-ln%285%2F48%29%2F5
or approximately,
k=-0.4524
So,
T-20=48e%5E%28-0.4524t%29
Find t when T=21.
21-20=48e%5E%28-0.4524t%29
e%5E%28-0.4524t%29=1%2F48
-0.4524t=ln%281%2F48%29%7B%7B%7B%0D%0A%7B%7B%7Bt=8.557
or
t=9hrs to the nearest hour.