You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on

Name: Problem 11, Chapter 1: Calculus Early Transcendentals
Uploaded: 2020-06-10T05:05:26-08:00
Duration: 9 min 6 s
Channel: Bruno Toledo
Description: You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.

You put some ice cubes in a glass, fill the glass with cold...

Key Concepts

Thermal Equilibrium

Thermal equilibrium refers to the state reached when objects in contact exchange heat until they settle at a uniform temperature. In this scenario, heat is transferred between the ice, the water, and eventually the surrounding air, leading the system to eventually settle to a single temperature value.

Phase Change and Latent Heat

During phase changes, such as the melting of ice, energy is absorbed without changing the temperature of the substance. This absorbed energy, known as latent heat, explains why, despite heat transfer occurring, the water’s temperature remains nearly constant until all the ice has melted.

Heat Transfer Mechanisms

Heat transfer in this context occurs through processes like conduction and convection. These mechanisms govern how energy is exchanged between the ice, water, and the environment, dictating the gradual changes observed in the system's temperature over time.

Transcript

00:01 Hello everyone, bruno toledo here, and today we're going to be answering the question you see on the screen.

00:07 So here's the situation.

00:09 We have cold water is poured into a glass containing ice cubes.

00:13 We need to show how the temperature of the water changes over time by sketching a rough graph of the temperature of the water as a function of elapsed time.

00:22 Okay, so we know the temperature, and we're going to let temperature here just to label our variables be big.

00:30 T capital t so capital t is going to be the temperature of the water the water inside the glass and we're going to let little t this is my lowercase t here that's going to be time that's going to be elapsed time so from the moment we pour the water right you start measuring the time so we know that the temperature is to be a function, let's call this function f.

01:16 It's going to be a function of this time, elapsed time.

01:23 Okay, so let's start labeling our axes so that we can start drawing the graph.

01:31 But before we do that, let's talk about what we have going on here.

01:36 Time cannot be negative in this situation, right? elaps time is going to start at t -equal zero, which is going to indicate the time, we finish pouring the water into the glass, and then it's going to move forward.

01:50 So t cannot be negative here.

01:54 It has to be non -negative.

01:56 And t is going to be our independent variable.

01:59 So that's going to be the x.

02:01 So little t is going to replace the x axis here.

02:06 The temperature, let's think about this.

02:10 The water is never really going to freeze, right? we're assuming that this glass is in a room that's a room temperature right it's not inside a freezer or anything like that so the water temperature is not really going to dip below zero degrees fahrenheit or celsius it doesn't matter which scale you're using it's not going to be a negative temperature right just pouring cold water in the glass containing ice cubes is not going to freeze the liquid in the glass.

02:46 So we also know that the temperature cannot be negative here, right? so that means big t has to be greater than or equal to zero, and little t has to be greater than or equal to zero.

02:58 So that means we're all we only need the first quadrant to graph what's going on here.

03:05 So let's go ahead and label some axes.

03:09 So let's draw our axes.

03:11 Again, we only really need the first quadrant.

03:25 So this axis here is going to be, let me use a different color.

03:35 Let's go with purple.

03:37 This axis here is going to be the temperature axis.

03:42 And this axis here is going to represent the time, elapsed time axis.

03:49 That looks ugly.

03:51 Let me fix that.

03:54 Little t.

03:56 Lowercase t.