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## Linear models

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# Linear equations word problems: volcano

CCSS.Math:

## Video transcript

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.