Key features of graphs — Harder example

- [Instructor] The graph above in the ta-plane approximates the Apollo 8's altitude, a, in kilometers, t seconds after lift-off. Approximately how many kilometers per second, how many kilometers per second was the Apollo 8 traveling between 156 and 184 seconds after lift-off? All right, so let's look at this chart. They're calling the ta-plane because our horizontal axis is the t-axis measuring time. Our vertical axis is the a-axis measuring altitude in kilometers. This is time in seconds. This is altitude in kilometers. Now, they wanted to know, well, what was the, how many kilometers per second was Apollo 8 traveling between 156 and 184 seconds? And they label those points for us. This right over here. This right over here, this point tells us, what was the altitude of Apollo 8 at 156 seconds? You see the 156 seconds right over here. So our t-coordinate, I guess you could say, is 156. And this right over here tells us the altitude at 184 seconds. It's 96 kilometers. At 156 seconds, it's 70 kilometers. At 184 seconds, it is 96 kilometers. So if we wanna find, if we wanna approximate how many kilometers per second was the Apollo 8 traveling between those two points, well, kilometers per second. We're figuring out the rate of change of altitude. And we should really be talking about kilometers per second in the vertical direction, is I guess what we should assume if we wanna be a little bit more precise. But that would be just the rate of change in altitude or the rate of change in altitude with respect to time. And we can approximate that by finding the slope of the line between these two points. So what's the slope of that line? It's gonna be change in altitude over change in time. So let's see, change in altitude is going to be 184 kilometers. And I can even keep it in kilometers. Write the units down. Make it a little clearer. Minus 156 kilometers. And then our change in time is going to be 96 seconds minus 70 seconds. So what's that going to be? 184 minus 156 is, it's about 20, let's see, if this was 154, it would be 30, so this is gonna be 28 kilometers. It's going to be 28 kilometers over, what is this? 26 seconds. Over 26 seconds. So this is going to be a little bit more, so slightly more. So 28, this is gonna be 28 over 26 kilometers per second. So this value right over here, this is a little bit more than one. One would be 26 over 26. So this is, you can view this as one and 2/26, or you can write this as one and 1/13. So a little bit more than one kilometer per second. So let's look at the choices right over here. And you look at the choices here, and only one of 'em is a little bit more than one. This is less than one, so you rule that out. This is a lot more than one, lot more than one. So we would go with that one right there.