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Applications of Linear Equations - Problem 3
To write an equation representing a word problem, start by defining what the variables, x and y, will stand for. "x" should represent the independent variable in the problem, and "y" should represent the dependent variable in the problem. In other words, the value of x will determine the value of y. If x changes, y will change accordingly. The value of y depends on the value of x. Set up an equation using x and y that represents the information in the word problem. Graph the equation by using a table of values or by plotting the y-intercept then using the slope to find other points along the line. Use the graph to find the x- and y-intercept, which is where the line crosses the x- and y-axes.
A scuba diver is 400 feet below sea level and raising at a rate of 4 feet every 3 seconds. The first thing we have to do is write an equation to represent the depth of the diver, letting y equal 0 represent the water surface.
So the first thing that I know, is that this guy starts out below the surface, below by 400 feet. That means I’m going to start with -400. That’s where he starts from. Then he’s raising up, he’s coming up, so I’m going to be adding 4 feet every 3 seconds. I’m going to use X represents the number of seconds.
Once I have my equation ready to go, I’m ready for part B which says sketch a graph. So I’m got exactly sure what my scaling is going to be yet. I could make a table but I’m going to try using y equals mx plus b strategies, and also the intercepts to help me get a good accurate graph.
Well, the first thing I know, is that I need to go, at least down to -400 for my Y values. So my Y axis is going to have to be pretty long, so that I can go into a negative area or quadrant 4. X, I’m only going to do in the positive direction because X represents time and we only have positive time in the real world.
So if this is the diver's depth in feet. He starts at -400. I’m just going to choose to have each box represent 20. So I have 20, 40, 60, 80, -100, -200. One, two, three, four, -300, oops! Wait, I’m going to try that again because I scaled incorrectly. This happens all the time. You guys, a lot of times on your graph paper, it’s so important to use pencil, because you do the mistake I just made. I ran out of space. So instead of having each box be 20, I’m going to say each box is 25, that way I can go, 100, 200, come on, I hope it fits, there’s 300, 400.
So you guys even brilliant mathematicians like me make these kinds of scaling errors. Just be really careful that before you get too committed to your graph, you have it properly scaled.
So I know he starts at -400 feet. The next thing I’m going to have to do is figure out how to represent my slope of +4/3. Before I can do that, I need to put some scaling and numbers on my X axis. So let’s see, I’m just going to pick to do the same scale, you don't have to. But I’m going to choose to say again that each box represents 25. One, two, three, four, 200, 300, and I’m going to stop there because I run oat of space. I might need to increase my graph, we’ll see.
So starting at -400, I’m going to go up 4 over 3. And this will work for my y equals mx plus b strategies, because I have the same scale on both the X and the Y axis, meaning I skipped by the same number of lines. Like, a box here was 25 on this axis and a box is 25, on that axis as well. So let’s go through, and using my y equals mx plus b techniques I’m moving up 4/3; one, two, three, four, over 3. I have a pretty good looking line I think.
That represents the diver as he’s coming up to the surface. One thing before I move on, I noticed that I forgot to label my X axis. Please, please, please especially when you have a real world situation, make sure you keep everything labelled so you can keep track of how Y represents feet, X represents seconds.
So when the question asks me to sketch a graph, I’m all set. I use y equals mx plus b. Part c says; find and interpret the X and Y intercepts. Well you know the Y intercept is where the graph crosses the Y axis. There it is right there, let me mark it in a different color. This is my Y intercepts right here. My Y intercept is (0, -400). That’s where the diver is right now.
Now we need to find my X intercept where my line crosses the X axis, there it is right there. It’s the point (300, 0). Think about what that means. He used to be 400 feet below the surface, at this point, he gets to 0 feet below surface, or he hits air. So when I’m asked to interpret the X intercept, I’m going to say this is where the diver reaches the surface or when the diver reaches the surface.
And this is really useful in real life because, if you were on the boat, and you were waiting for that diver guy to come up, and you couldn’t contact him but you knew how fast he was moving, you would know that he should get to the top after about 300 seconds.
This is the kind of situation where a real world problem can be modelled using a graph. And you can do it precisely using a ruler to help you and using the X and Y intercepts, you can make some really good extrapolations to the real world.