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Constant of proportionality

by autodidude
Tags: constant, proportionality
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autodidude
#1
Oct11-11, 01:36 AM
P: 333
If a ∝ b and a ∝ c, why do you multiply b and c together to find the constant?

I also noticed something, but am not sure of the reason why. If you find the constants individually for each expression and combine them all, you get a the the power of the number of expressions

e.g.

a ∝ b
a ∝ c
a ∝ d

So the individual constants would be say, k1, k2 and k3 respectively. If you then multiply it all together

a ∝ (k1b)(k2c)(k3d)

You get a^3

If there're expressions, then a^4 etc. All of the numbers I've tried so far have yielded the result but I'm not sure why that's happening

Thanks
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mathman
#2
Oct11-11, 03:19 PM
Sci Advisor
P: 6,037
Your question is very confusing.
Mute
#3
Oct11-11, 04:12 PM
HW Helper
P: 1,391
Quote Quote by autodidude View Post
If a ∝ b and a ∝ c, why do you multiply b and c together to find the constant?
You don't. If [itex]a\propto b[/itex] and [itex]a \propto c[/itex], then that tells you that [itex]a = k_3 bc[/itex], where k3 is some constant of proportionality. This is because:

Given both [itex]a = k_1(c)b[/itex], where k1(c) is a proportionality factor that you know depends on c, and [itex]a = k_2(b)c[/itex], where k2(b) is a proportionality factor that depends on b, you can divide the two equations to get

[tex]1 = \frac{k_1(c)b}{k_2(b)c},[/tex]

or

[tex]\frac{k_1(c)}{c} = \frac{k_2(b)}{b}.[/tex]

However, by assumption k1 depends only on c and k2 depends only on b, so the only way this relation can hold is if both sides are equal to the same constant, say k3. It follows then that [itex]a = k_3 bc[/itex].

I also noticed something, but am not sure of the reason why. If you find the constants individually for each expression and combine them all, you get a the the power of the number of expressions

e.g.

a ∝ b
a ∝ c
a ∝ d

So the individual constants would be say, k1, k2 and k3 respectively. If you then multiply it all together

a ∝ (k1b)(k2c)(k3d)

You get a^3

If there're expressions, then a^4 etc. All of the numbers I've tried so far have yielded the result but I'm not sure why that's happening

Thanks
I'm not sure what you're talking about here. If a is proportional to all those variables, then [itex]a = k_4bcd[/itex], by similar logic to what I did above. I'm not sure where these powers of a comes from.


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