3-D graph problem

[peacefreak77]

1 pt) Six functions are displayed below. All six of these are functions of three variables (x,y,z), so they are functions of a point's location in three-dimensional space. One of these functions could be written as a function of only the distance from the y axis: This function will return the same value for all points (x,y,z) that are the same distance from the y axis. Which function is this?

Which of the following functions can be expressed in terms of the distance from the y axis? (a, b, and c are constants.) Warning: You will only be given 3 attempts at this problem!


A. f(x,y,z) = 1/\sqrt{x^2 + b^2 + z^2}
B. f(x,y,z)=\sin(x^2+y^2+z^2)
C. f(x,y,z) = 2 x + 3 z + c
D. f(x,y,z) = \ln(3 y^2)
E. f(x,y,z) = 10 e^{y+a}
F. f(x,y,z)=b + x^2 z^2

okay. this is from my multi-variable calc course, but as it really doesn't require calculus, just genuine thought (which i have been incapable of lately), i think it's okay to post it here.

WORK: i know that the y axis has coordinates of (0,y,0). so if for instance i was looking for all points that were 5 units away from the y axis, i could start with the points (0,y,5), and all values of y would make that five units from the y axis. also the points (5,y,0) are all 5 units from the axis. also, i know there exist points that are five units from the axis that have non-zero x and z values, but i have been too confused to think about that a lot.
i have been able to eliminate all answers that have a y in them, since the y varies and can change the output. i haven't figured out how to eliminate between a,c, and f though.

helpful- the distance from the y axis is given by squroot((0-x)^2+(y1-y)^2+(0-z)^2)

any help would be greatly appreciated!

 

[pka]

If \(\displaystyle (p,q,r)\) is a point then its distance from the y-axis is \(\displaystyle \sqrt {p^2 + r^2 }\).
Your notation is hard to read. However, from the fact above the answer should be clear.

 

[peacefreak77]

where did you get that formula from? i'm still confused.

yes, i admit my notation is difficult to read. i'm sorry. i copied it off of our internet homework system.

 

[pka]

where did you get that formula from?

Suppose that \(\displaystyle l(t) = P + Dt\) is a line and \(\displaystyle X\) is a point then the distance from \(\displaystyle X\) to \(\displaystyle l(t)\) is \(\displaystyle \frac{{\left\| {\vec {PX} \times D} \right\|}}{{\left\| D \right\|}}.\)
Thinking of the y-axis as a line \(\displaystyle jt\) then \(\displaystyle \left\| { < p,q,r > \times j} \right\| = \left\| { - ri + pk} \right\| = \sqrt {r^2 + p^2 }.\)
There you have it.

 

[Gene]

Or... The points a given distance from the y axis form a circular cylinder of that given radius. That's the formula of that cylinder.

 

[peacefreak77]

but if the points the given distance from the y axis form a cylinder/sphere/etc. then the output should be the same from all of those points on the cylinder/sphere/etc., i mean f(x,y,z,) should be constant for all of those points.
how can i prove that?

 

[stapel]

...the points the given distance from the y axis form a cylinder/sphere/etc.

A cylinder is not the same as a sphere. A cylinder (or tube, if you will) with the y-axis as its central axis, will always be a fixed distance (the radius) from the y-axis. A sphere (a hollow ball) will either cross/touch the y-axis, or it won't, but its distance from the axis will not be uniform.

Eliz.

 

[pka]

Suppose that \(\displaystyle (p,q,r)\) is a constant distance from the y-axis, so that \(\displaystyle {r^2 + p^2 }\) is constant.
Therefore, would \(\displaystyle \sqrt {r^2 + b^2 + p^2 }\) be constant?

 

[peacefreak77]

yes, i know there is a difference between a cylinder and a sphere, but thank you for the explanation, that did help me. is option A a cylinder? i can't visualize 3D space very well.

pka, yes, i think i realized that that was true, making A correct, but why are the other two options with no y value not constant? why is p^2+r^2 constant while p^2*r^2 isn't? that's where i get lost.

 

[pka]

why is p^2+r^2 constant while p^2*r^2 isn't? that's where i get lost.

In option F is \(\displaystyle p^2 r^2\) a constant.
But how is that about distance?
Do you know about the standard metric?

 

[peacefreak77]

no, i don't.

i think i am confusing the distance formula with the answer options. but if the points are all a constant distance away then won't all of them result in the same output if you plug them into choice F?

 

[peacefreak77]

nevermind, it just all made sense. A involves the formula for a circle. if the figure described by A is a sphere, then the radius is constant no matter the x and y values so the z value must be constant.

i didn't articulate that well. but i think i understand.

thank you very much for the help.