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Applications of Linear Equations - Problem 2
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, so 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 using a table of values. Choose a few x-values and plug them into the equation in order to find the corresponding y-values. Plot the coordinate points to draw the line of the equation. Use the graph to find different point values along the line.
This is a kind of tricky word problem that has to do with a linear equation, because not only do you have a negative slope, but you also have a fractional slope. Here’s what I mean.
A hot air balloon is 200 feet up and descending at a rate of 10 feet every 3 minutes. Write an equation to represent the height of the balloon and the function of time. Let’s do that piece first before we move on.
So my equation is going to start with height equals, starts at 200, from there it's coming down. So I’m going to use a negative sign, and it comes down 10 feet every 3 minutes. X is going to represent my number of minutes Y represents my height in this function.
For part B they want me to make a graph. But before I do that, I’m going to make a table using some values to help me get some good points for my graph. It will also help me figure out my scaling, like how far I should be going on my X and Y axis.
So I’m going to show you a trick. Since my X value is being multiplied by 10/3, I’m going to choose to use X numbers that are multiples of 3, that way I won’t get fractional numbers and the 3 will cancel out. So if I use 0, I’m going to pick, 12, 30 and 60. I just chose those X numbers because they’re multiples of 3 and when I substitute them in there, those 3 multiples on top will cancel out with that 3.
I’m going to grab my calculator but I can do the first one without it. Y equals 200 take away 10/3 times 0 is going to be 200. 200 take away 10/3 times 12, be careful with the order of operations. Do 10 times 12 first before you divide by 3 and then after that add 200. You get 160.
Then I’m going to do -10 times 30 divided by 3 plus 200 from my next one. I already did this ahead of time, and I got 100 there. Here I got 0. So those table values are going to help me draw the graph. I know that my X scale needs to go all the way out to 60. So the next trick is figuring out how many boxes I should skip by.
Looking at your graph paper, sometimes you can count how many are on your page, and divide it in a clever way so you can get the 60 boxes. I’m just going to kind of guess. I’m going to say, okay, each box is going to represent 3, let’s do 6. And the reason why I’m using 6 is so I can be sure it fits on my page.
So going across, I’m going to have 6, 12, 24, 36, 48. Let me write those a little bit bigger for you. 12, 24, 36, 48, hopefully you can see that better. My next value would be 60, so that each box represents 6, but then I’m only writing the number for every 12. You’re allowed to do that. Just make sure you’re skipping by the same number of boxes each time you use 6.
That’s going to be my X which represents number of minutes, and now I’m ready to do my Y. Y represents height of the balloon and I need to go all the way up to 200. So in order to get all the way up to 200, again, I want to think about how much each box should represent. I’m going to say each box is 10. So I have 10, 20, 30, 40, 50. I’m just feeling like I’m going to mark every 50, 60, 70, 80, 90, 100, got to get all the way up to 200. 150, 200. Let me just double-check that. Okay good. And this represents my height of the balloon in feet. Height in feet, oops! kind of run out of space there.
Next thing I’m going to do is get my dots on the grid; (0, 200), (12, 160), (30, 100). 30 is not on here so I’m going to have to remember that each box is 6. 30 goes to 100 and then 60 is 0. Use your ruler to connect them and make sure your line is ruler straight, otherwise you made an error either in your scaling or in your table. There we go.
So that line represents 200 take away 10/3x and it also represents how high the balloon is. That was all part B. Here comes part C. These are going to be a lot quicker now that I have a good graph. Part C, how high will the balloon be after 30 minutes? This is why a good graph is super useful. I can just look at my thing and say, “Oh 30 minutes? The balloon is 100 feet high.”
Another way I could have found that out is by looking at my table. I chose, just kind of I was lucky, I chose 30 as one of my X values and there it is 100 feet high. Last but not least, how long until the balloon reaches the ground?
Well look at your graph. Vertically we’re seeing how high the balloon is, it starts at 200 feet, it’s coming down, coming down, coming down. It hits the ground right there after 60 minutes or one hour depending on how you want to write it.
So if you guys can get yourselves to where you can draw accurate graphs, it takes a long time and there is a lot of places to make errors. But if you can get good graphs, it makes the subsequent questions, when you’re interpreting your graphs, go much, much, much quicker.