A roasted turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 76°F.
(a) If the temperature of the turkey is 155°F after half an hour, what is its temperature after 45 min? (Round your answer to the nearest whole number.)
______°F
(b) After how many hours will the turkey cool to 100°F? (Round your answer to one decimal place.)
________hr
Use dT/dt = k(T-T0)
Where T is the temperature of the turkey, T0 is the temperature of the surroundings and k is a constant.
"Rate of change of temperature is proportional to difference between temperature of turkey and surroundings"
-Separate and integrate
-Use the provided info(155F after 1/2h) to find constant k
-Sub in values to get answer
To answer these questions, we need to understand the concept of heat transfer and how temperature changes over time.
(a) Let's calculate the rate of temperature change for the turkey first. We know that the temperature of the turkey decreases with time as it cools down in the room. The rate at which the temperature changes is directly proportional to the temperature difference between the turkey and its surroundings (in this case, the room).
The rate of temperature change can be represented by the equation:
dT/dt = -k(T - T_room)
Where dT/dt is the rate of change of temperature with respect to time, T is the temperature of the turkey, T_room is the temperature of the room, and k is a constant that depends on the properties of the turkey and the surroundings.
We can rearrange the equation to solve for dT:
dT = -k(T - T_room)dt
Now we can integrate both sides to find the relationship between temperature and time. Assuming k remains constant during the cooling process:
∫dT/(T - T_room) = -k∫dt
The left side of the equation can be integrated using logarithms:
ln|T - T_room| = -kt + C
Where C is the constant of integration.
Now, let's use the given values. After half an hour (30 minutes), the temperature of the turkey is 155°F, and the room temperature is 76°F. Substituting these values into the equation:
ln|155 - 76| = -k(30) + C
ln|79| = -30k + C
At this point, we don't know the value of k or C, but we can solve for the difference in temperature using the initial condition provided:
ln|185 - 76| = -k(0) + C
ln|109| = C
Now we have both C and ln|79| with unknown constants. Let's subtract the two equations:
ln|79| - ln|109| = -30k
Taking the exponential of both sides to eliminate the logarithm:
e^(ln|79| - ln|109|) = e^(-30k)
79/109 = e^(-30k)
Now we can solve for k:
k = -ln(79/109)/30 ≈ 0.0109 (approximately)
Now that we have the value of k, we can determine the temperature of the turkey after 45 minutes. Let's substitute t = 45 into the original equation:
ln|T - 76| = -0.0109(45) + ln|109|
Now solving for T:
T - 76 = e^(-0.0109(45) + ln|109|)
T = e^(-0.0109(45) + ln|109|) + 76
T ≈ 143°F (rounded to the nearest whole number)
So, after 45 minutes, the temperature of the turkey is approximately 143°F.
(b) Now, let's find out how long it takes for the turkey to cool to 100°F. We can use the same equation and solve for time (t):
ln|T - 76| = -0.0109t + ln|109|
Substituting T = 100 and solving for t:
ln|100 - 76| = -0.0109t + ln|109|
ln|24| = -0.0109t + ln|109|
Now, isolate t:
-0.0109t = ln|24| - ln|109|
t = (ln|24| - ln|109|)/(-0.0109) ≈ 2.5305 hours (rounded to one decimal place)
So, the turkey will cool to 100°F after approximately 2.5 hours.