Newton's Law of Cooling.
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 75°F.
(a) If the temperature of the turkey is 145°F after half an hour, what is its temperature after 45 min? (Round your answer to the nearest whole number.)
(b) After how many hours will the turkey cool to 100°F? (Round your answer to one decimal place.)
drag out your formula for Newton's Law of Cooling, and plug in your numbers.
What do you get?
To answer these questions, we can use Newton's Law of Cooling, which states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature. The formula for Newton's Law of Cooling is:
dT/dt = -k(T - Ta)
where dT/dt is the rate of change of temperature with respect to time, T is the temperature of the object at any given time, Ta is the ambient temperature, and k is a constant.
a) To find the temperature of the turkey after 45 minutes, we can first find the value of the constant k using the initial condition:
dT/dt = -k(T - Ta)
Plugging in the initial conditions T = 185°F, Ta = 75°F, and t = 0 (since it's the initial condition), we get:
dT/dt = -k(185 - 75)
dT/dt = -k(110)
Now, we use the fact that the temperature of the turkey is 145°F after half an hour (30 minutes) to find the value of the constant k:
dT/dt = -k(110) = (145 - 75)/(30 - 0)
-110k = 70/30
k = -70/(30 * 110)
Simplifying, we get:
k = -7/330
Now that we have the value of k, we can find the temperature of the turkey after 45 minutes:
dT/dt = -k(145 - 75)
dT/dt = -(-7/330)(145 - 75)
dT/dt ≈ -(-1.318)(145 - 75)
dT/dt ≈ -(-1.318)(70)
dT/dt ≈ -92.26
To find the temperature after 45 minutes, we need to integrate this equation:
∫ dT/dt dt = ∫ -92.26 dt
T = -92.26t + C
Applying the boundary condition T(0) = 185°F, we find that C = 185°F. Now, plug in t = 45 minutes (or 0.75 hours):
T = -92.26(0.75) + 185
T ≈ -69.195 + 185
T ≈ 115.805
Rounding to the nearest whole number, the temperature of the turkey after 45 minutes is 116°F.
b) To find the time it takes for the turkey to cool to 100°F, we can set up the equation:
dT/dt = -k(T - Ta)
Plugging in T = 100°F, Ta = 75°F, and solving for t:
-dT/dt = -k(T - Ta)
-dT = -k(T - Ta) dt
Integrating both sides:
∫ -dT = ∫ -k(T - Ta) dt
-T = -kt + C
Applying the boundary condition T(0) = 185°F, we find that C = -185. Now, plug in T = 100°F:
-100 = -kt - 185
kt = 85
t = 85/k
Substituting the value of k we found previously (-7/330):
t ≈ 85/(-7/330)
t ≈ -85 * 330/7
t ≈ -330 * 85/7
t ≈ -330 * 12.143
t ≈ -3976.02
Taking the absolute value, the time it takes to cool to 100°F is approximately 3976.02 minutes.
Converting to hours, t ≈ 3976.02 minutes / 60 minutes/hour ≈ 66.27 hours.
Rounding to one decimal place, it takes approximately 66.3 hours for the turkey to cool to 100°F.
To solve both parts of the problem, we can use Newton's Law of Cooling, which states that the rate of change of the temperature of an object is directly proportional to the difference between the object's temperature and the ambient temperature.
The formula for Newton's Law of Cooling is:
dT/dt = -k(T - Ta)
where dT/dt represents the rate of change of the temperature with respect to time, T is the temperature of the object, Ta is the ambient temperature, and k is a constant that depends on the specific properties of the object and its surroundings.
Let's use this formula to solve both parts of the problem.
(a) To find the temperature of the turkey after 45 minutes, we need to first find the value of k. We can use the initial and final temperatures of the turkey, the ambient temperature, and the time.
The initial temperature (To) of the turkey is 185°F, the final temperature (T) is 145°F, the ambient temperature (Ta) is 75°F, and the time (t) is 30 minutes.
dT/dt = -k(T - Ta)
Now we substitute the given values into the equation:
(145 - 75) = -k(185 - 75)
Simplifying this equation gives us:
70 = 110k
To find k, divide both sides by 110:
k = 70/110
k ≈ 0.6364
Now that we have the value of k, we can use it to find the temperature of the turkey after 45 minutes. Again, using the formula:
dT/dt = -k(T - Ta)
(145 - Ta) = -0.6364(185 - Ta)
Expanding and simplifying the equation:
145 - Ta = -0.6364 * 185 + 0.6364 * Ta
Now solve for Ta:
Ta ≈ 150.2°F
Since we need to round our answer to the nearest whole number, the temperature of the turkey after 45 minutes is approximately 150°F.
(b) To find out how many hours it will take for the turkey to cool to 100°F, we need to find the time (t) when the temperature (T) of the turkey is 100°F. We can use the formula:
dT/dt = -k(T - Ta)
The initial temperature (To) of the turkey is 185°F, the ambient temperature (Ta) is 75°F, and the final temperature (T) is 100°F.
dT/dt = -k(T - Ta)
dT/dt = -0.6364(100 - 75)
Now, we rearrange the equation:
dT/dt = -0.6364(25) / t
Since dT/dt means the rate of change of temperature with respect to time, we can rewrite it as:
dT/dt = ΔT / Δt
So the equation becomes:
ΔT / Δt = -0.6364(25) / t
Simplifying, we get:
Δt / ΔT = t / -0.0255
Now, re-arranging the equation:
t = -0.0255 * Δt / ΔT
We have Δt = 60 minutes (since we need to find the time in hours) and ΔT = 185°F - 100°F = 85°F.
Substituting the values in:
t = -0.0255 * 60 / 85
t ≈ -0.018
Since time cannot be negative, we discard the negative sign:
t ≈ 0.018 hours
Rounding the answer to one decimal place, the turkey will cool to 100°F after approximately 0.02 hours.
Therefore, the turkey will cool to 100°F after roughly 0.1 hours or 6 minutes.