Cooling Time of a Pizza Pan A pizza pan is removed at 5: 00...

Cooling Time of a Pizza Pan A pizza pan is removed at 5: 00 PM from an oven whose temperature is fixed at $450^{\circ} \mathrm{F}$ into a room that is a constant $70^{\circ} \mathrm{F}$. After 5 minutes, the temperature of the pan is $300^{\circ} \mathrm{F}$. (a) At what time is the temperature of the pan $135^{\circ} \mathrm{F}$ ? (b) Determine the time that needs to elapse before the temperature of the pan is $160^{\circ} \mathrm{F}$. (c) What do you notice about the temperature as time passes?

Cooling Time of a Pizza Pan A pizza pan is removed at 5: 00 PM from an oven whose temperature is fixed at $450^{\circ} \mathrm{F}$ into a room that is a constant $70^{\circ} \mathrm{F}$. After 5 minutes, the temperature of the pan is $300^{\circ} \mathrm{F}$.
(a) At what time is the temperature of the pan $135^{\circ} \mathrm{F}$ ?
(b) Determine the time that needs to elapse before the temperature of the pan is $160^{\circ} \mathrm{F}$.
(c) What do you notice about the temperature as time passes?

Key Concepts

Differential Equation Modeling

The cooling process can be modeled using a first-order differential equation, typically written as dT/dt = k(T - T_ambient), where k is a constant. Solving this differential equation leads to an explicit formula for the temperature, which is essential for predicting the cooling time at various temperatures.

Asymptotic Behavior

As time goes on, the temperature of the cooling object approaches the ambient temperature, meaning that although the temperature continuously decreases (or increases if coming from a lower value), it never actually reaches the ambient temperature in finite time. This concept highlights the long-term behavior of systems described by exponential decay.

Exponential Decay

Exponential decay describes a process where the quantity—here, the temperature difference between the object and the ambient temperature—decreases at a rate proportional to its current value. In the context of cooling, the temperature drop follows an exponential curve as time progresses.

Newton's Law of Cooling

This principle states that the rate at which an object cools (or warms) is proportional to the difference between its own temperature and the ambient temperature. It forms the basis for modeling cooling processes and leads to an exponential temperature decay towards the environment’s temperature.

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