Is this relationship directly proportional?

[bonbon22]

Homework Statement

If i had an equation y=3x^(2/3) would this relationship be directly proportional? as y/x^(2/3) = 3 so the gradient is constant

if thats the case would y =3x^2 be directly proportional also ??

Relevant Equations

no equations

I was confused by the question below

http://www.antonine-education.co.uk/Pages/Physics_4/Tests/Test%20on%20Fields.pdf
if you look at question at question 1 section B same reasoning is used to answer this question as you can see the graph is a straight line??
but plotting r against T ^2/3 or any two variables shouldn't give you a straight line?

 

[fresh_42]

If you introduce a new variable $u=x^{\frac{2}{3}}$ then $y \sim u$ is directly proportional, but this is not the case for $y$ and $x$. The same is true for $y \sim x^2$.

In the first case we have $y^3=3x^2$ so $y^3$ and $x^2$ are directly proportional.

 

[bonbon22]

If you introduce a new variable $u=x^{\frac{2}{3}}$ then $y \sim u$ is directly proportional, but this is not the case for $y$ and $x$. The same is true for $y \sim x^2$.

In the first case we have $y^3=3x^2$ so $y^3$ and $x^2$ are directly proportional.

if you check out the link , the graph does not seem to add a new variable yet the line is directly proportional.
also does directly proportional not mean the graph has to be straight line going throuugh the origin so y^3 = 3x^2 on a graphing calculator online is not a straight line?

 

[Mark44]

if you check out the link , the graph does not seem to add a new variable yet the line is directly proportional.

A line can't be "directly proportional." When you have two quantities A and B that are proportional, it means that their ratio is constant. IOW, that $\frac A B = k$, where k is a constant. Another way to say this is that A = Bk.

also does directly proportional not mean the graph has to be straight line going throuugh the origin so y^3 = 3x^2 on a graphing calculator online is not a straight line?

They are not saying that y is proportional to x. The equation you wrote is equivalent to $y = 3^{1/3}x^{2/3}$. This means that y is proportional to $x^{2/3} = \sqrt[3]{x^2}$. Here the constant of proportionality is $\sqrt[3] 3$.

 

[bonbon22]

A line can't be "directly proportional." When you have two quantities A and B that are proportional, it means that their ratio is constant. IOW, that $\frac A B = k$, where k is a constant. Another way to say this is that A = Bk.
They are not saying that y is proportional to x. The equation you wrote is equivalent to $y = 3^{1/3}x^{2/3}$. This means that y is proportional to $x^{2/3} = \sqrt[3]{x^2}$. Here the constant of proportionality is $\sqrt[3] 3$.

i see now cheers m8

 

[bonbon22]

A line can't be "directly proportional." When you have two quantities A and B that are proportional, it means that their ratio is constant. IOW, that $\frac A B = k$, where k is a constant. Another way to say this is that A = Bk.
They are not saying that y is proportional to x. The equation you wrote is equivalent to $y = 3^{1/3}x^{2/3}$. This means that y is proportional to $x^{2/3} = \sqrt[3]{x^2}$. Here the constant of proportionality is $\sqrt[3] 3$.

one question m8 my equation was y = 3x^(2/3) is this the same as y= 3 * 3 sqaure root ( x^2) ??
how come you cube rooted the 3 also ??

 

[fresh_42]

one question m8 my equation was y = 3x^(2/3) is this the same as y= 3 * 3 sqaure root ( x^2) ??
how come you cube rooted the 3 also ??

$a^{\frac{1}{n}} = \sqrt[n]{a}$. Just take this to the power of $n$ to see it is the same. Thus $3\sqrt[3]{x^2}=3\sqrt[3]{x}^2 =3x^{\frac{2}{3}}$.

 

[bonbon22]

$a^{\frac{1}{n}} = \sqrt[n]{a}$. Just take this to the power of $n$ to see it is the same. Thus $3\sqrt[3]{x^2}=3\sqrt[3]{x}^2 =3x^{\frac{2}{3}}$.

mark 44 said that that equation 3x^(2/3) is the same as 3 root (3) * 3root ( x^2)
bit confused on that . thanks for the reply tho.

 

[fresh_42]

mark 44 said that that equation 3x^(2/3) is the same as 3 root (3) * 3root ( x^2)
bit confused on that . thanks for the reply tho.

He wrote

A line can't be "directly proportional." When you have two quantities A and B that are proportional, it means that their ratio is constant. IOW, that $\frac A B = k$, where k is a constant. Another way to say this is that A = Bk.
They are not saying that y is proportional to x. The equation you wrote is equivalent to $y = 3^{1/3}x^{2/3}$. This means that y is proportional to $x^{2/3} = \sqrt[3]{x^2}$. Here the constant of proportionality is $\sqrt[3] 3$.

which means the constant here wasn't three - sorry, but your inline formulas are hard to decode - but $3^{\frac{1}{3}}= \sqrt[3]{3}$ which is a special case of what I wrote with $a=n=3$. Hence $y=\sqrt[3]{3}x^{\frac{2}{3}} \sim x^{\frac{2}{3}}$ with proportionality factor $3^{\frac{1}{3}}$.