A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the

A warm can of soda is placed in a cold refrigerator. Sketch...

Key Concepts

Exponential Decay

The temperature of an object cooling in a constant environment typically follows an exponential decay curve. This means that the rate of temperature change is fastest at the beginning and asymptotically approaches the ambient temperature, resulting in a curve that is steep at first and then levels off as the object reaches thermal equilibrium.

Newton's Law of Cooling

This concept explains how the rate of temperature change of an object is proportional to the difference between its own temperature and that of the surrounding environment. It provides the mathematical framework, using a differential equation, to model the cooling (or heating) process, illustrating that a larger temperature difference results in a more rapid change initially, and as the difference decreases, the rate slows down.

Thermal Equilibrium

This is the state that the object eventually reaches where its temperature becomes equal to the ambient temperature. At thermal equilibrium, the net heat transfer ceases, and the temperature of the object remains constant. In the cooling process, the changing rates of temperature adjustment in the graph approach zero as equilibrium is reached.

Transcript

00:01 This is problem number forty five of this to a calculus eighth edition section two point seven a warm can of soda is placed on in a cold refrigerator.

00:10 Sketch the graph of the temperature of the soda as a function of time.

00:14 So if we do this part first here we have some axes.

00:18 Capital tear presents the temperature of the soda and lower case t represents time and for reference, we will quit time as, ah, one hour here or just to have sixty minutes just ahead of you.

00:44 Some references, thirty minutes and events on.

00:55 We're going to sketch the graph of the temperature so there's going to be a rough sketch.

00:58 Let's say that it begins with a certain temperature.

01:02 A tea initial, a german rate as t in the showroom.

01:14 If it starts at this temperature and it's placed any cold refrigerator, its temperature will begin to decrease, and it will decrease to a point where it finally reaches the temperature of the refrigerator, which will do know as t final.

01:30 So if i show here just a dotted line, you can take it.

01:35 This is a horizontal asking, too, because thie soda will come to equilibrium with the temperature of the refrigerator eventually, um, but at first, since it's at a higher temperature than the all right refrigerator, it'll sharply decrease first and then begin to level off because what we should, or what we would interpret as the gradual decrease of the temperature of this, they cannot sodium after its place into a cold refrigerator...

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