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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: EspoirTron on May 22, 2013, 09:26:23 pm

Title: constant of proportionality
Post by: EspoirTron on May 22, 2013, 09:26:23 pm: Could someone please care to explain what this concept is and whether or not we need to no it for Math Methods?
Title: Re: constant of proportionality
Post by: b^3 on May 24, 2013, 12:38:51 am: Quote from: kezzab on May 22, 2013, 09:26:23 pm
Could someone please care to explain what this concept is and whether or not we need to no it for Math Methods?
What context is this in?
For example if we say that the gradient of a curve is proportional to say the $x$ value, that is $\begin{alignedat}{1}\frac{dy}{dx} & \propto x\end{alignedat}$ . What this is saying is as one changes, the other will change and they will be related by a constant. Then to express the gradient in terms of $x$ , we would have $\begin{alignedat}{1}\frac{dy}{dx} & =kx\end{alignedat}$ which is where our constant of proportionality comes in.

I'm know sure if thats answered your question, but hope it helps. Note: You could also say have $y$ is proportional to $x$ , which would be $y=kx$ .
Title: Re: constant of proportionality
Post by: b^3 on May 24, 2013, 01:04:21 am: Well yeh, extensions of the simple example were meant to be inferred.

The only example that comes to mind for methods is say, when you have exponential growth. Say we have $N$ , the number of bacteria in a culture, which follows $N=N_{0}e^{kt}$ where $N_{0}$ is the initial number of bacteria. Then we can show that the rate at which the number of bacteria increases will be proportional to the number of bacteria in the same at that time. That is
$\begin{alignedat}{1}N & =N_{0}e^{kt} \\ \frac{dN}{dt} & =kN_{0}e^{kt} \\ \frac{dN}{dt} & =kN \\ \implies\frac{dN}{dt} & \propto N \end{alignedat} $
Title: Re: constant of proportionality
Post by: BubbleWrapMan on May 24, 2013, 12:26:14 pm: The term 'constant of proportionality' was used in the 2011 Methods exam 2. So you apparently need to know what it means. I think this was a surprise to most people that year, though.
Title: Re: constant of proportionality
Post by: EspoirTron on May 24, 2013, 05:42:02 pm: Quote from: Timmeh on May 24, 2013, 12:26:14 pm
The term 'constant of proportionality' was used in the 2011 Methods exam 2. So you apparently need to know what it means. I think this was a surprise to most people that year, though.

That was what I was talking about. I did that question and initially the concept through me off. However, after going over it with my teacher we didn't have to solve for k or even know about it (it's exact value for calculating T). Despite this, the concept threw me off. Does anyone know if there have been similar questions on trial exams?
Title: Re: constant of proportionality
Post by: BubbleWrapMan on May 24, 2013, 07:21:37 pm: I wouldn't know about trial exams, but I can explain it a bit further if it helps.

Think of it like a linear relationship, without the ' $+c$ '. So something like $y=mx$ . As an example, you might not know the exact relationship between $y$ and $x$ , but you may know that if you double $x$ , then $y$ doubles, and if $x$ triples then $y$ triples, et cetera; they always increase or decrease together by the same factor. But since you don't know the exact relationship, you just write $y=mx$ .

You could also rearrange that to $\frac{y}{x}=m$ . This is more intuitive, since it just says that 'the ratio of $y$ and $x$ is always the same', which is another way of saying that $y$ is proportional to $x$ . Of course, this has the slight disadvantage of causing division-by-zero issues, so we generally write $y=mx$ .

In order to work out the value of a proportionality constant, you need extra information. In the VCAA question, these conditions were given in various ways to allow you to determine $k$ on different days. In our example of $y=mx$ , you'd simply need a point that the line passes through other than the origin. It's really just an unknown, so you need one equation to find it, in general.
Title: Re: constant of proportionality
Post by: EspoirTron on May 24, 2013, 09:02:26 pm: Quote from: Timmeh on May 24, 2013, 07:21:37 pm
I wouldn't know about trial exams, but I can explain it a bit further if it helps.

Think of it like a linear relationship, without the ' $+c$ '. So something like $y=mx$ . As an example, you might not know the exact relationship between $y$ and $x$ , but you may know that if you double $x$ , then $y$ doubles, and if $x$ triples then $y$ triples, et cetera; they always increase or decrease together by the same factor. But since you don't know the exact relationship, you just write $y=mx$ .

You could also rearrange that to $\frac{y}{x}=m$ . This is more intuitive, since it just says that 'the ratio of $y$ and $x$ is always the same', which is another way of saying that $y$ is proportional to $x$ . Of course, this has the slight disadvantage of causing division-by-zero issues, so we generally write $y=mx$ .

In order to work out the value of a proportionality constant, you need extra information. In the VCAA question, these conditions were given in various ways to allow you to determine $k$ on different days. In our example of $y=mx$ , you'd simply need a point that the line passes through other than the origin. It's really just an unknown, so you need one equation to find it, in general.

Ah yes I see then, thank you very much for that it helped me a lot!