Beg Algebra: Inverse Proportion

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INVERSE PROPORTION
Okay, a wonderful, in fact really powerful application of inversely proportional, or inverse proportionality, actually is with light. You know, you take light for granted. You know, light is just light, it goes and whatever. But in fact, light is really, really an amazing object or an amazing thing to study, and physicists study this all the time. And in fact, the reason why you can see me is because of light, and the passing of light, and so forth. But if you just think of sort of raw light, not just a beam of light, well that sort of––there’s the beam of light. That light actually is traveling, and the illumination that is given, of course, drops as you go further and further from the subject; the light actually drops. And it turns out––and this is real––that it drops inversely proportional to the square. So the illumination produced by a light source varies inversely as the square of the distance from that source. So think about it for a second: The further you are from a light source, the less bright the light looks; and the closer you are to the light source, the more bright the light looks. And it turns out those things are inversely proportional to the square. So in fact, if you know a certain illumination, so you have an illumination, that’s going to be inversely proportional to the distance squared.
So how do I write that down? I would say that the illumination––I’ll call that “I” for illumination––is inversely proportional, so 1 over the distance squared ( 21Id=). So the further away you are, as this number gets bigger (distance), this whole thing gets smaller, so illumination gets smaller. The closer you become, so then this number (distance) actually gets smaller, then this whole fraction gets larger, the more illumination, and in fact this law actually captures the spirit of what happens with light. So what this means is “I” equals some constant divided by the distance squared (2kId=).
Now suppose that for this light source, for example, at 5 meters the illumination is 70 candela––that’s the way you measure illumination. So that means that when we have a distance of 5 meters, the illumination is 70 candela. Well, I can then find the constant because I would say that 70 would equal this constant divided by 5 squared, which is 25 (2705k=), and so what I would see here is that the constant (k) would just be 25 times 70. And you can multiply that out, in fact if you want me to I can do that really fast for you: 25 times 70 equals 1750. And so what that means is that for this particular light source––of course different light sources will have different constants––but for this particular light source, the illumination is equal to 1750 all divided by d squared (21750Id=). So now knowing this, here’s a question: What is the illumination at a distance of 12 meters? Well, how would I find that out? Well, what I would do is I would put in 12 for the distance and see what the illumination is. And so that illumination would equal 1750 divided by 12 squared, which is 144 (1750144I=). And so what would that equal? Let’s try that really fast. So I take 1750 and divide it by 144, and that would equal around 12.1527… candela. So in fact that’s the illumination. So you can see that it’s a dramatic drop from the given information, which at 5 meters we had 70 candela, and then at 12 meters, we’re actually going to have 12.15, a dramatic decrease in the illumination as you go further and further out.
Okay, so that takes care of inverse proportionality, in fact this is inverse squared proportionality, and all from light. So this is a real-world example. And of course this begs the bonus question, so now it’s time for that bonus question, and the bonus question of course is: Where were you when the lights went out? I’ll see you at the next lecture.
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