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This is the problem that I am having trouble with for my test review. I am completely blank and don't know what it is asking for. Can you please guide me step by step. For example: Why did constant $k$ appear all of a sudden?

$a$ varies directly with $b$

Which of these equations could represent the relationship between $a$ and $b$?

$a$ varies directly with $b$ if $a=k\cdot b$ for some constant $k$

If you divide each side of this expression by $b$, you get $\displaystyle\frac ab=k$ for some constant $k$.

$\displaystyle\frac ab=\frac12$ fits this pattern, with $k=\displaystyle\frac12$

  • $a=\frac12-b$
  • $\frac12\cdot\frac1a=b$
  • $2\cdot\frac1a=b$
  • $\frac12\cdot a=\frac1b$
  • $\frac ab=\frac12$
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You might want to explain where radiation enters into this. Also there's no constant $K$ in your image. I suspect you mean $k$? – joriki May 3 '12 at 15:28
it doesn't i'm sorry edited it Ok well I don't know how to put the k – Backtrack May 3 '12 at 15:28
How do you mean, you don't know how to put it? You just put it in the comment; what keeps you from putting it in the question in the same way? – joriki May 3 '12 at 15:34
Ok man that doesn't really matter. That's not even the point. Can you please help me solve it step by step I would really appreciate it – Backtrack May 3 '12 at 15:36
Well, I'm not sure why I should take time to answer your question if you think it doesn't really matter whether you put in the minimal effort of making your question correspond to what you're asking about; but I did. – joriki May 3 '12 at 15:43
up vote 0 down vote accepted

"$a$ varies directly with $b$" basically says "if you double $b$, then $a$ is doubled; if you multiply $b$ by $10$, then $a$ gets multiplied by $10$", etc.; the two are in direct proportion to each other. The constant $k$ is the proportionality factor. For instance, if the values $1,2,3,4$ for $b$ correspond to values $2,4,6,8$ for $a$, then the proportionality constant is $k=2$, whereas if the values $1,2,3,4$ for $b$ correspond to values $5,10,15,20$ for $a$, then the proportionality constant is $k=5$. I suggest that you play around with the equations offered as choices on the right and convince yourself that the last one is the only one that has the above property. The explanation on the lower left explains that in terms of formulas.

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You see when I read " "a" varies directly with "b" " I thought that it meant that a and b have nothing to do with each other. The reason I though that, was because it said vary, as in different – Backtrack May 3 '12 at 15:50
@Backtrack: I see. It's actually the opposite; "vary with" (whether directly or not) means that if one is varied, the other (usually) also varies. – joriki May 3 '12 at 15:51
Oh ok so basically "a varies directly with b" would mean. " Whatever you do to a you have to do to b" Also, what if it just said "a varies with b" without "directly" What would be the case there? – Backtrack May 3 '12 at 16:02
@Backtrack: Yes, "Whatever you do to $a$ you have to do to $b$", but in a certain sense, namely multiplicatively. (Adding to $a$ whatever you add to $b$ could also be described as "doing to $a$ whatever you do to $b$", but that wouldn't be called "varying directly with".) Without "directly", the expression "vary with" is very general and covers any form of dependence, e.g. temperature varies with time, the volume of a sphere varies with its radius, etc.; no particular form of dependence is implied in that case. – joriki May 3 '12 at 16:10
Ok thank you very much. And sorry for getting mad at first, it's just that I was really frustrated I appreciate your help – Backtrack May 3 '12 at 16:18

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