Key features of graphs — Harder example

- [Instructor] Ruby puts a dish of macaroni and cheese into the regrigerator. The dish was 80, 80 degrees Fahrenheit when she put it in. And the inside of the refrigerator was 40 degrees Fahrenheit. The dish cooled quickly at first, so this is gonna be interesting, quickly at first, then slowed as it approached the temperature inside the refrigerator, all right. Which of the following graphs in the mT-plane could best represent the temperature T, in degrees Fahrenheit, m minutes after Ruby put the dish in the refrigerator? So mT-plane sounds very fancy, but all they're saying is is the horizontal axis, that's our time axis, and it's gonna be measured in minutes, m for minutes. And our vertical axis in this plane, in this coordinate plane, is going to be our temperature axis, and it's measured in Fahrenheit degrees. So let's see what they're saying. So the dish started at 80, at 80 degrees, at 80 degrees Fahrenheit. So at zero minutes, right when she put it into the fridge, it should be 80 degrees. So this one right over here, this one is indeed at 80 degrees. This graph right over here, this is starting at zero degrees, not at, it needs to start at 80 degrees if we're gonna model what actually happened. So this, we can rule that one out immediately. Then when you look at these choices, this one's starting at 40 degrees. Once again, not what we need. We need it to start at 80 degrees at zero minutes, so we'll rule that one out. This one also starts at 80 degrees. Now let's think about the shape of what the curve should be. They say the dish cooled quickly at first, then slowed, then slowed, as it approached the temperature inside the refrigerator. Now, does this graph describe that? It starts, before it was in the refrigerator it was at 80 degrees, or right when they put it in the refrigerator, it's at 80 degrees. And then it starts, its temperature declines quickly at first, but then the rate of decline slows as it approaches the temperature inside of the fridge, that 40 degrees, and that seems like that this is what's happening. It's becoming, the slope is becoming, it starts out quite negative, but it becomes less and less and less negative as we get closer and closer to the 40 degrees Fahrenheit. This one over here, after some number of minutes, we do end up at this 40 degrees Fahrenheit. But the rate of decline is constant the entire time. This is a line, this is describing a constant rate of decline. But what they described is a changing rate of decline. The dish cooled quickly at first. So that would be a steep decline, cooled quickly at first, then slowed as it approached the temperature inside the refrigerator. So the decline slows, it becomes less negative as we approach the temperature of the refrigerator. So you have this change in the rate of temperature change. And so you are going to, or at least I would, pick this one right over here. This one right over here describes a constant rate, which is not what was described in the question.