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

1. Sep 10, 2007

### dt_

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

1. The problem statement, all variables and given/known data

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

2. Relevant equations

x=3y is the given line.

3. 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!! :)

2. Sep 10, 2007

### Dick

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|?

3. Sep 10, 2007

### dt_

$$| \frac{x}{3} | ^{2} = \frac{x^{2}}{9}$$

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?

4. Sep 11, 2007

### Dick

That looks good to me.

5. Sep 11, 2007

### HallsofIvy

Staff Emeritus
A cone, in the mathematical sense, has two parts. Each one is a "nappe" of the cone.

6. Sep 11, 2007

### dt_

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?

7. Sep 11, 2007

### Dick

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.