Question

A pork roast was taken out of a hardwood smoker when its internal temperature had reached 180°F and it was allowed to rest in a 75°F house for 20 minutes after which its internal temperature had dropped to 170°F. Assuming that the temperature of the roast follows Newton’s Law of Cooling, a) Express the temperature of the roast, T as a function of time t. 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 1

The student's question is about applying Newton's Law of Cooling to find an expression for the temperature of a pork roast as a function of time and to determine when the roast cools to a specific temperature.

Step-by-step explanation:

The question involves an application of Newton's Law of Cooling to express the temperature of a pork roast as a function of time. To find this expression, we utilize the formula for Newton's Law of Cooling: T(t) = T_s + (T_0 - T_s) * e^(-kt), where T(t) is the temperature at time t, T_s is the surrounding temperature, T_0 is the initial temperature, and k is a constant that represents the cooling rate.

Assuming the temperature of the roast follows Newton’s Law of Cooling, for part a), as we know the roast's temperature drops from 180°F to 170°F in 20 minutes in a 75°F environment:

1. Let T_s = 75°F be the room temperature (surrounding temperature).
2. Let T(0) = 180°F be the initial temperature of the roast when taken out of the smoker.
3. Let T(20) = 170°F be the temperature of the roast after 20 minutes of resting.
4. Let k be the cooling constant we need to solve for.

For part b), to find when the roast would drop to 140°F, we solve T(t) = 140°F for t using the previously determined function.