Beg Algebra: Direct Proportion

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DIRECT PROPORTION
A fantastic real world example of two things being proportional to each other actually can be found in slinkies. They’re almost hypnotic. You know, I think this will lower your blood pressure. Maybe before a test, you can just do a slinky kind of a… Anyway, the thing about a slinky is that it is basically an example of a spring and you’ll notice with a spring if you put weight on something, in fact, the spring comes down. And how much force you put on it is proportional to the amount of weight you put on it. And in fact, this is known as Hook’s Law.
So let’s take a look at this question and see if we can answer it using Hook’s Law. So this actually a real law by the way. Don’t break this law whatever you do. So Hook’s Law says that the, for a slinky or more generally for any elastic spring, states that the distance that a slinky stretches varies directly as the force applied. So let me actually show you this as an example. So here’s a little mini slinky. This is a slinky for like the underprivileged kids. I can barely even do this. But look what happens. If I take this and I put a little weight on it, notice that the distance the slinky comes down is so much. If I put more weight on it, this big one, then notice that the slinky drops a lot more. And it turns out that the force put upon the slinky is proportional, directly proportional to the distance that the slinky stretches. So there’s a direct proportionality.
So what I want to do now is to analyze that with the following question. The question is—let me actually do it with this example. And it’s written over there. You can read it. If I put a force of 15 pounds. So let’s suppose this is 15 pounds, then I'm told that the slinky stretches eight inches. This is just a mock up to give you a sense of this. So I put 15 pounds, I have the slinky stretching eight inches. Now the question is just knowing that and Hook’s Law, which says that those things are proportional, the question is how far will the slinky stretch if I put 30 pounds on. So I want to find this distance now using Hook’s Law. Well, the way to proceed is just to think and know the fact that the force is actually proportional to the distance. So what I have is the following. If d is the distance that the spring is stretched, so it’s this distance right here. Then that’s proportional to the force, which in this case would be the weight put upon it.
Now what does it mean for two things to be directly proportional? It means that d is equal to some constant times f. So the question is what’s that constant, what’s that number that actually makes these things related. Well, actually that changes from spring to spring depending upon how tight the spring is or how loose the spring is. You see, that constant is actually depending upon the actual spring itself.
So let’s see what it would be in this case. Well, we actually have a data point so we can actually find the constant. Because we know if we put a 15 force on the spring, it’s going to stretch eight inches. So that’s given to us. So that will allow us to identify the constant. So, in fact, if I put in 8 here and 15 here, I know this has to hold because with a force of 15 pounds, I know it stretches eight inches. This allows me to solve for k. So what I see is that constant is 815. Is that the answer to the question? Absolutely not. That’s just the constant that allows me now to figure out exactly how the stretching of the string depends upon the force. And so what I now see, coming back to here, is that d—you see, before I just knew that the things were equal up to a constant, but now I know exactly numerically what the constant is. It’s 815x f. So now I can answer how far does the slinky get stretched if I were to put on a 30-pound weight. Well, no problem. I just put in a force of 30 here. So 815x 30; I can cancel a little bit. The 15 and the 30, this becomes just a 2 and 8 x 2 is 16. So it would actually stretch 16 inches. So you see how I did that? Just knowing their proportional, with one piece of information I was able to find the constant, then go back to the proportionality and make this equality and then actually answer the question. So, in fact, that allowed me to actually figure out that when I come up and put on that big weight, this thing is going to stretch 16 inches. Pretty cool and actually true. A real world example just using a slinky.
Up next I'll take a look at an example that uses the inversely proportional law which again is real, real world. I'll see you there.
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