Linear equations word problems: volcano

Zane is a dangerous fellow who likes to go rock climbing inside an active volcano. He is a dangerous fellow. He just heard some rumblings, so he's decided to climb out as quickly as he can. Zane's elevation relative to the edge of the inside of the volcano in meters, E, as a function of time in seconds is shown in the table below. Zane climbs at a constant rate. So this guy, I mean if we were to draw a volcano here, this guy is just kind of silly. So this is my volcano. And he's actually climbing on the inside of an active volcano. So there's probably smoke and ash and all the other stuff coming out of this thing. So this really is dangerous for him. And let's say that this right over here is Zane. He's climbing up from inside the active volcano. So let's think about what they're telling us. So based on the table, which of these statements is true? So I'm not going to even look at these statements here. I'm just going to try to interpret this. So his elevation as a function of time in seconds is shown in the table below. So his elevation is negative 24 when time is equal to 0. And this table is done in a kind of nontraditional way. Normally, we would have the input into the function on the left-hand side. And then we would have the function of it on the right-hand side. And actually I like looking at things that way, so I'm going to make it like that. So let me copy and paste this so I can put it on the other side. So let me cut and let me paste it, paste it right over here. So this one, now I can think of it a little bit clearer. So at time 0, he's going to be at negative 24 meters. At time 4 seconds, he's going to be at negative 21 meters. So this makes a little bit clearer, at least in my head. So let's think about what's happening. So where does he start? At time equals 0, where is he? Well at time equals 0, he is 24 meters below the edge of the volcano. So this distance at time equals 0, this distance right over here is 24 meters. And we could even plot this in a graph. So this is his elevation relative to the edge, and it is a function of time. I'll write it like that. And it is negative most of this time. So I'm going to make the t-axis a little bit higher. So it looks something like that. That's our t-axis. And when t is equal to 0, we see that his elevation is negative 24 meters. So his elevation is negative 24 meters. So he's going to, this is right here at 0 seconds. And then when time increases by 4, so our change in time is equal to 4, what's his change in elevation? Well, his change in elevation is, let's see, he's going from negative 24 to negative 21. He increased by 3. So his change in elevation is equal to positive 3. He increased by 3. So at what rate is he increasing his elevation with respect to time? Well, change in elevation is equal to 3 per unit. And that's 3 when his change in time. And remember this triangle just means a Greek letter delta, shorthand for change in. So change in elevation over change in times is 3 over 4. So one way to think about this is that he goes 3/4 of a meter per second. The units up here is meter. The units down here is second. So he goes 3/4 of a meter per second. And we can verify that. The next row here, we see our change in time is 8. So it's twice as much time has passed, so he should have gone twice as much distance if his rate is constant. Let's verify that that's the case. So he went from negative 21 to negative 15. His elevation increased by 6. So change in elevation over change in time is 6/8, which is the same thing as 3/4. So you see that he has this constant change. So let's plot a few of these points. So when time is 0, his elevation is negative 24. When time is 4, right over there, his elevation is negative 21. Let's say this looks something like this. And so his elevation as a function of time is going to look something like this. Let me actually draw it a little bit more to scale. Because the other thing that we do know is that when time is 32, his elevation is 0. So let me put that right over there. When time is 32, his elevation is 0. So his elevation as a function of time looks something like this. And we could plot other points there when time is 4. So 4 is going to be at this half. That's a 4. So 4 is going to be right over there. His elevation is negative 21. So this is a general idea. He starts at negative 24 meters and he increases at a rate of 3/4 meters per second. So which of these choices is correct? Zane was 24 meters below the edge of the volcano when he decided to leave, and he climbs 3 meters every 4 seconds on the way out. That seems right. He climbs 3 meters every 4 seconds. So we're going to go with that one. Let's make sure that these aren't right. Zane was 24 meters below the volcano when he decided to leave, and he climbs 4 meters every 3 seconds. No, no, it's 3 meters every 4 seconds. So that's not right. Zane was 32 meters below the edge of the volcano. No, that's not right. Zane was 32 meters. That's not right either.