What $3$-dimensional shape is represented by graph of the set of pints $ (rcos\theta, rsin\theta,z)$ where $r$ is a constant real value, $\theta$ range from $0$ to $2\pi$ radiant and $z$ range over the real number.
I was thinking that the graph might be a cylindrical polar coordinate system, but when I tried to use a graphing app it seems like Elliptic cone. Would someone explain to me and show me a graph of it.
On
These are cylindrical coordinates. Look it up!
Notice that if we set $z=c=constant$, these coordinates define a circumference of radius $r$ at height $z=c$. Now take this circumference and change $z$ from $c$ to any other point of the $z$-axis. This defines a cylinder.
I suspect that what you did to obtain an elliptic cone was to take the position vector in this coordinate system $\mathbf{x}=(r\cos\theta,r\sin\theta,z)$ and change the angle $\theta$ from $0$ to $2\pi$. Of course, for a given $r$, this defines a cone, but you forgot that there is another variable, $z$, which may change as well, effectively defining a cylinder.
Hint: If you fix $z$ to equal some real number $C$, notice that you restrict the image to the plane $z = C.$
Then, what is the image of the set of points when this $z$ is fixed? i.e., what does $(r \cos \theta, r \sin \theta, C)$ look like when $\theta$ ranges between $0$ and $2\pi$?