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# Direct variation models

Video transcript

Let's do some word problems that
are essentially dealing with slope of a line. You might see these being
referred to as direct variation models, because we're
going to model what's being described in
this problem. We're going to graph it out and
then hopefully we're going to be able actually answer
their question. Let's see what they're asking. The current standard for
low-flow showerheads is 2.5 gallons per minute. Calculate how long it would
take to fill a 30 gallon bathtub using such a showerhead to supply the water. What we can do here is we can
set up a direct variation model, which sounds very fancy,
but it just says, OK, let's set up a little equation
that describes how many gallons we would have filled--
or how many gallons we would have used after a certain
number of minutes. So let's say that we have
gallons is equal to the rate at which we fill the gallons. So it's going to be 2.5 gallons
per minute times the number of minutes. I'll say m for minutes
and g for gallons. We have just set up our direct
variation model. Nothing fancier than that. We now have an equation that
describes, you give me the number of minutes, I'll multiply
that times 2.5 because that's how quickly we're
filling the bathtub. So after 1 minute, 1 times
2.5, we have 2.5 gallons. After 2 minutes, we have 2 times
2.5, we have 5 gallons. So this is our model. And this is also a line. Remember the form of a line is
y is equal to mx plus b. Here we have no b. The b is gone. We just have an m times an x. The x now we're calling minutes
and the y we're now calling gallons and the
slope is now 2.5. Let's plot this before I even
answer their question. Instead of calling this the
x-axis-- remember x, the independent variable,
is now the minutes that we let it flow. So that's the m-axis
for minutes. And the vertical axis, instead
of calling it the y-axis, I'm going to call it the g-axis
for the number of gallons we've filled. I'm only going to deal in the
positive quadrant assuming we can only have positive
minutes. So what's going on here? We have a slope of 2.5. We could also write this as
gallons is equal to-- 2.5 is the same thing as 5/2 gallons
per minute times the minutes. So now we know our
slope is 5/2. I could have used 2.5 as
well, but I like 5/2. Our y-intercept we already
know is 0. There's no y-intercept here. You could do this as plus 0. So you start over here
at the origin. That's our y-intercept. And for every 2 that we go to
the right, we move up 5. So we move-- change in x is 2. One, two, three, four, five. Change in x is 2. One, two, three, four, five. If change in x is negative 2,
then change in y, it'll be negative 5. Negative 2. One, two, three, four, five. And so on and so forth. We'll eventually
get down here. So our line, our model if we
graph it, looks like this. I'm trying my best to go through
all of the points. Actually, I just said
I would only do it in the first quadrant. It really doesn't make sense in
this quadrant right here, because you can't have
negative minutes. Or you shouldn't have
negative minutes. So we should only be dealing
with it up here. Now they're asking how long
will it take to fill a 30-gallon bathtub? Now, unfortunately, my graph
right here does not go all the way up to 30 gallons. If it did-- this is 10 gallons
right up here. That's 10 gallons if I went 3
times higher, I could just read the graph. But we could also solve for it
algebraically right here. How many minutes does it take? Well, let's just make our
gallons equal to 30. So you have 30 gallons is equal
to, I'll put it in the same color, 2.5 gallons per
minute times minutes. Now, all we have to do to solve
for minutes is divide both sides by 2.5 gallons
per minute. Divide both sides by 2.5
gallons per minute. I'm doing the units to show you
that the units actually all work out in the end. So this will cancel out. It will just become a 1. And so the left-hand side-- or
we could say m-- is going to be equal to 30 divided by 2.5. We have this gallons
in the numerator. I want to show you, you can deal
with the units just like you would deal with
actual numbers. And if I have gallons per minute
in the denominator, if I divide by this fraction,
that's the same thing is multiplying by its inverse. That's the same thing
as multiplying by minutes per gallon. Right? These units were in
the denominator. When I put them in the
numerator, I flip them. So gallons in the numerator,
gallons in the denominator. They cancel out. So I'm left with 30 divided
by 2.5 minutes. And what is 30 divided by 2.5? Is equal to 12. So it'll take us 12 minutes to
fill up a 30 gallon bathtub. We have our minutes
right there. 12 minutes. Let's do one more. Amen is-- or maybe A-man, I
don't know the best way to pronounce that name-- is using
a hose-- let me scroll over a little bit-- is using a hose to
fill his new swimming pool for the first time. He starts the hose at 10 PM--
let me write this down. He starts at 10 PM-- so this is
the start time-- and leaves it running all night. At 6 AM, he measures the
depth and calculates the pool is 4/7 full. So when he starts at 10 PM--
so this is time and this is how full the pool is. So, obviously when he starts
it, the pool is empty. It's a new swimming pool. They tell us. So the pool is 0 full. It's 0 whatever. It has no water in
it whatsoever. Then at 6 AM, he measures the
depth and calculates the pool is 4/7 full. So here it is 4/7 full. At what time will his
new pool be full? So we want to know
when it's full. When it is 1/1 full? Where it's 7/7 full. At what time? So to do this, we have to set up
a similar model that we did the last time. We could say, the fullness of
the pool is equal to some constant times the amount of
time that's passed by. We know when time is equal to
0-- let me put this this way. This is time. Let me write it here. This is time. This is time, 0. What is this in hours? This is 8 hours later, right? This is time 8. We don't know what this is. This is time something else. So when time is 0 at 10 PM, 0
times k, we have 0 fullness. We are not full at all. When time is equal to 8, we have
k times 8. k is the rate at which we're filling
the pool. k times 8. We're at 4/7 fullness. So now we can actually
figure out what k is. We can figure out what our
proportionality constant is for our direct variation
model. Sounds very fancy, but all we're
saying is, look, this pool-filling business
can be modeled by an equation like this. The amount that we're full is
directly proportional to the amount of time that we
let the hose run. And this is the proportionality
constant. We don't know how quickly it
fills, but now we can figure out how quickly it fills. Because we know after 8
hours it is 4/7 full. So to solve for k, you divide
both sides by 8 hours. So we get k is equal to 4/7 full
divided by 8 hours, which is the same thing as 4/7 times
1/8 full per hour. So if we figure this
out, let's see. Divide by 4. Divide by 4. So we get 1/14 fullness. It's kind of a weird unit--
full per hour. Or you could say, we fill 1/14
of the pool per hour. So k is 1/14. So here our equation-- let
me write over here. The fullness of the pool is
equal to 1/14 times the time. So the question that we
have to answer is when does this equal 1? At what time? So let's set up the equation. So we have 1. That means we're completely
full. It is equal to 1/14
times time. If we multiply both sides of
this equation times 14, the 1/14 and the 14 cancel out. And we're left with
t is equal to 14. So the pool will fill
after 14 hours. Remember everything we were
dealing with was hours. If we start at 10 PM-- that was
time 0-- what time are we at 14 hours? So 10 PM in one day-- if you
go to 10 AM the next day, that's 12 hours. That's how I think. We need to go 2 more hours
to get to 14 hours. That's noon the next day. At noon the day after he starts
filling is when the pool will be full. We can graph this. I had this graph paper here. Let's actually graph everything
I'm talking about. The equation-- I wrote it over
here-- fullness is equal to 1/14 times t. Let's assume that each of
these notches is two. That is two, four, six, eight,
ten, twelve, fourteen. So this tells us that as
we run 14, we rise 1. So if x-change is 14, y-change
is positive 1. And I'll make these
units 1, 2. So the scale isn't
exactly perfect. I'm distorting the graph a
little bit, but that is 1. So the graph will look
something like that. Just like that. It has a slope of 1/14. Anyway, hopefully you
found that useful.