Proportionality

Agent Smith said:
🤔 You mean they belong to the same category but are nonidentical? I get that for your examples \(\displaystyle y = -x\) and \(\displaystyle y = -2x\), but what about \(\displaystyle y = -x\) and \(\displaystyle yx = 1\) in my question?
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I am confused. Is the idea of things having similar attributes but not being the same new to you?
 
Agent Smith said:
You mean to say then, per our agreed upon definition, that this \(\displaystyle y = -x\) expresses a direct proportionality relationship? Perhaps we should exclude \(\displaystyle y = -ax\) from our set of relationships (this was from a science forum and I haven't encountered negatives, except maybe as the charge on an electron, in physics formulae).
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Agent Smith said:
So I was on a science forum and saw an interesting question. ... This confused the poster, as does it me, because this is how inverse proportionality \(\displaystyle y \propto \frac{1}{x}\) is also described.

How would you to clear this matter up?
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How about showing us exactly what was said that you are asking about, rather than make everyone guess?
 
Agent Smith said:
I would if I could, but my Google's malfunctioning
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Translation: I can't because I made it up. I hope this excuse works so the thread will die off, but then I'll start on another topic and pretend this never happened.

On to the next thread... and the cycle repeats.
 
mmm4444bot said:
Hello. Going forward, please provide links to or images of any source material to be discussed when starting new threads. Thank you!
[imath]\;[/imath]
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It's on another forum and I don't have permission from the author to use his material. I've presented the gist of what he had an issue with in the OP.

I'll give some examples so we have something to work with

1. A car fuels up to max capacity (50 liters of gasoline). It consumes 1 liter of gasoline for every 10 km traveled.
L = Fuel left in the gas tank. d = Distance traveled. \(\displaystyle F = 50 - \frac{d}{10}\)

gas.PNG

2. Boyle's law: Pressure of a gas is inversely proportional to its volume i.e. [imath]P \propto \frac{1}{V} \implies PV = k[/imath] (a constant)
Let k = 2 and so [imath]PV = 2[/imath]
boyle.PNG


Combining the 2 graphs we get ...
boyle2.png

What are some valid conclusions we can draw from the above?
 
Agent Smith said:
What are some valid conclusions we can draw from the above?
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What conclusions do you draw? I see nothing of interest here at all. It certain says nothing about proportions. And the graphs can't be combined, because their variables and units are entirely different.
 
Dr.Peterson said:
What conclusions do you draw? I see nothing of interest here at all. It certain says nothing about proportions. And the graphs can't be combined, because their variables and units are entirely different.
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We can superimpose the graphs (as in the last, 3rd, image). Notice, in general, for both curves
1. The slope at any point on the curve is negative. The description, as x increases y decreases applies.
2. For lack of a better word, I'd say the mechanism (the functions) that produces the effect described above differs. [imath]y = -mx^1 + b \text{ vs. } y = kx^{-1}[/imath]
3. In some sense I'd have to agree that [imath]y = -mx + b \overset{??} \equiv yx = k[/imath]
 
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I also wonder why some laws in nature that I know of are restricted to
1. [imath]y = kx[/imath] (Direct proportionality)
2. [imath]yx = k[/imath] (Inverse proportionality)

I kinda understand 1 as we must start from [imath]0[/imath], that's how I understand it any way.
What about 2? We can never attain perfect vacuum (for Boyle's law)?

And [imath]y = kx[/imath] is NOT the inverse of [imath]yx = k[/imath], correct?
 
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