A pork roast was taken out of a hardwood smoker when its internal temperature reached 180°F. It was allowed to rest in a 75°F house for 20 minutes, after which its internal temperature dropped to 170°F. Assuming that the temperature of the roast follows Newton’s Law of Cooling:
a) Express the temperature of the roast, [tex]T[/tex], as a function of time [tex]t[/tex].
b) Find the amount of time that has passed when the roast would have dropped to 140°F had it not been carved and eaten.
Answer (3)
To express the temperature of the pork roast, T, as a function of time t, we can use Newton's Law of Cooling. According to the law, the rate of change of temperature is proportional to the difference between the temperature of the object and the surrounding temperature.
Taking the temperature of the roast at any given time as T(t) and the surrounding temperature as Ts, we can write the differential equation as:
dT/dt = -k(T - Ts)
Where k is the cooling constant.
Solving this differential equation will give us the temperature of the roast as a function of time.
To find the amount of time it would take for the roast to drop to 140°F, we need to solve the differential equation for t, when T(t) = 140°F, Ts = 75°F, and T(0) = 170°F.
To find the temperature T as function of time, use the formula T(t) = T_s + (T_0 - T_s)e^{-kt}. Given the initial condition and surrounding temperature, one can calculate the constant k and then solve for time t when the temperature reaches a particular value, such as 140°F. ;
Using Newton's Law of Cooling, we express the temperature of the pork roast as a function of time and use this function to find the time it would take for the roast to drop to 140°F. The temperature function is given by T ( t ) = 75 + 105 e − k t , and we can find t by solving for when T ( t ) = 140° F .
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