Surfaces

fifthrapiers

Consider the following surface:

x = 4y^2 + 4z^2

What's this surface called?

What coordinate plane does z=0 define?

What's the trace of this surface in the xz plane look like?

What's the trace of this surface in the planes x=k look like?

earboth

Consider the following surface:

x = 4y^2 + 4z^2

What's this surface called?
What coordinate plane does z=0 define?
What's the trace of this surface in the xz plane look like?
What's the trace of this surface in the planes x=k look like?
Hello,

I've attached a diagram of this surface.

a) It is a rotation paraboloid. The axis of rotation is the positive x-axis.

b) It's a parabola with the vertex in the origin: $\displaystyle x = 4y^2$

c) It's a circle: $\displaystyle k = 4x^2+4y^2 \Longleftrightarrow x^2+y^2=\left( \frac{1}{2} \sqrt{k} \right)^2$

EB

Attachments

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• fifthrapiers

Soroban

Hello, Ideasman!

I use the standard orientation of the three coordinate axes.
Code:
z
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* - - - - - - y
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x
Consider the following surface: .$\displaystyle x \:= \:4y^2 + 4z^2$

What's this surface called?
It is a paraboloid.

What coordinate plane does $\displaystyle z=0$ define?
$\displaystyle z = 0$ is the $\displaystyle xy$-plane (the "floor" of the graph).

What's the trace of this surface in the $\displaystyle xz$-plane look like?
The $\displaystyle xz$-plane is the "left wall" of the graph.

Let $\displaystyle y = 0$ and we have: .$\displaystyle x = 4z^2$

This is a parabola on the "left wall", vertex at the origin,
. . opening in the positive $\displaystyle x$-direction.

What's the trace of this surface in the planes $\displaystyle x=k$ look like?

Let $\displaystyle x = k$. .We have: .$\displaystyle 4y^2 + 4z^2\:=\:k\quad\Rightarrow\quad y^2 + z^2\:=\:\frac{k}{4}$

These are circles: centered on the $\displaystyle x$-axis with radius $\displaystyle \frac{\sqrt{k}}{2}$

• fifthrapiers