3D Surfaces - Equation Formed When Rotating a 2D Line About an Axis?

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Hi everyone, I'm pretty new to Physics Forums but it seems like a fairly friendly community. :)

Homework Statement



Determine the equation of the surface formed when the line x=3y is rotated about the x-axis.

Homework Equations



x=3y is the given line.

The Attempt at a Solution



First I write it in terms of x because it's simpler: y = \frac{1}{3} x

The slope is 1/3, thus, and you have a diagonal line that passes through the origin in a 2-D graph with the X-Y plane.

Now, if you rotate this about the X-axis, you see you get a sort of cone. rather, two cones, one for each side of the y-axis ; these two cones have their tops(tips) facing each other.

How, though, can I determine an equation for the cone? I know there is a generic equation that involves x,y,z variables and a,b,c constants (I think it's something like.. (x-a)^{2} + (y-b)^{2} = (z-c)^{2} )

but what do I plug in for the variables and constants? I think I need to substitute \frac{x}{3} for x , or maybe with one of the other variables (y or z) but I'm not sure where and how.

If anyone could help me on this I would very much appreciate it. Thank you! :)
 
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You are SOO dumb! Oh, wait, we're friendly! :) Just kidding. The absolute value of y at any point x determines the radius of the circle in the y-z plane with center at y=0, z=0, right? So what's the equation of such a circle in y-z with radius |x/3|?
 
| \frac{x}{3} | ^{2} = \frac{x^{2}}{9}

so the r^2 in x^2 + y^2 = r^2 is equal to (x^2) / 9

and..

the equation of the circle is
y^2 + z^2 = (x^2 / 9)


but that's a.. cone? i think?

which would make sense

does that seem right?
 
That looks good to me.
 
A cone, in the mathematical sense, has two parts. Each one is a "nappe" of the cone.
 
Thanks Dick. :)

HallsOfIvy: Not quite getting that, but do you think you can check my arithmetic and see if I've worked out the solution correctly?
 
I'm betting Halls already checked the math. He tends to catch small errors. As far as the "nappe" goes he's just telling that full geometry of your equation looks like a 'double cone' (one for x>0 and one for x<0), but it's still ok to call it a 'cone'. "nappe" is generic name for one of these two parts.
 
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