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?
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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 xaxis.
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
Hello, Ideasman!
I use the standard orientation of the three coordinate axes.Code:z




*       y
/
/
/
x
It is a paraboloid.Quote:
Consider the following surface: .$\displaystyle x \:= \:4y^2 + 4z^2$
What's this surface called?
$\displaystyle z = 0$ is the $\displaystyle xy$plane (the "floor" of the graph).Quote:
What coordinate plane does $\displaystyle z=0$ define?
The $\displaystyle xz$plane is the "left wall" of the graph.Quote:
What's the trace of this surface in the $\displaystyle xz$plane look like?
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.
Quote:
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 $\displaystyle \frac{\sqrt{k}}{2}$