End behavior of algebraic models | Mathematics III | High School Math | Khan Academy

A barista poured a cup of coffee. The initial temperature of the coffee was 90 degrees Celsius. As time t increased, the temperature c of the coffee began to decrease exponentially and approach room temperature of 20 degrees Celsius.

Which of the following graphs could model this relationship?

So, we're starting at 90 degrees Celsius. It looks like all of the graphs start at 90 degrees Celsius at t equals zero, and we are going to get—we're going to approach the room temperature of 20 degrees Celsius.

So this first one does approach the room temperature of 20 degrees Celsius as t increases. Now this one, when t is 70—I'm assuming this is in minutes—when t is 70, it looks like it has the temperature going to zero degrees Celsius. So that cup of coffee is going to start freezing, so I think I could rule out B.

Also, this looks like a linear model, not an exponential one. C does get us to this end state that stays at 20 degrees, but it doesn't look like an exponential model. It looks like it's linearly decreasing, and then it stops linearly decreasing after 50 minutes, and then it just stays constant at that temperature of 20. So even though it gets us to the right place, it does not look like an exponential decay, so I would rule choice C out as well.

So A is looking good. D, we are starting at 90. It does look like an exponential function. We have exponential decay right over here, and we are approaching something, but it's not the room temperature of 20 degrees Celsius. We're approaching 30 degrees Celsius here, so I'd also rule out D.

So A is looking good. It's an exponential; it's decreasing exponentially, starting at 90 degrees Celsius, and it's approaching the room temperature of 20 degrees Celsius.

Let's do another one of these.

So it says—let me scroll up a little bit—so it says that after the closing of the mills, the population of the town starts decreasing exponentially. The graph below presents the population P, in thousands, of the town T years after the closing of the mill.

Alright, so it looks like the population starts at 40,000. It's decreasing exponentially. It looks like over time the population is approaching 20,000 people.

So what is the question here? Based on the graph with the mill closed, what does the population of the town approach as time increases?

Well, we just said it. As time increases, it looks like it's coming close to—it's approaching 20,000. It's approaching 20,000. It's already gotten below 22,000. As far as you know, it looks like by after twenty or twenty-two years, we've already gotten below 22,000.

So we're definitely below 30 or 40,000, but we haven't gotten below 20,000, but we are approaching it. And we can even check our answer if we like.