Direct variation models

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.