A 20-pound turkey that is at the room temperature of 72^∘ is placed in the oven at 1:00 pm. The

step 1 So this is Newton's Law of Cooling and 75 degrees is the ambient temperature. This is how long i

A 20-pound turkey that is at the room temperature of 72^∘ is...

A 20-pound turkey that is at the room temperature of $72^{\circ}$ is placed in the oven at 1:00 $\mathrm{pm}$. The temperature of the turkey is observed in 20 minute intervals to be $79^{\circ}, 88^{\circ}$, and $96^{\circ}$. A turkey is cooked when its temperature reaches $165^{\circ}$. How much longer do you need to wait until the turkey is done?

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Key Concepts

Exponential Temperature Model

The exponential temperature model uses an exponential function to describe how the temperature of an object changes over time. Because the rate of temperature change is proportional to the temperature difference, the object’s temperature approaches the ambient temperature asymptotically, following an exponential curve rather than changing linearly.

Time Constant

The time constant is a parameter that emerges from the exponential model in thermal processes, indicating the timescale over which noticeable temperature changes occur. It quantifies how rapidly the temperature of an object approaches equilibrium, playing a crucial role in predicting when a desired temperature will be reached in heating or cooling scenarios.

Newton’s Law of Cooling/Heating

Newton’s Law of Cooling/Heating explains that the rate at which an object’s temperature changes is proportional to the difference between its current temperature and the constant ambient temperature. This law is fundamental for modeling processes where an object moves toward thermal equilibrium with its surroundings, whether it is cooling down or warming up.

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