Question

Identify the surface for each of the following equations.
(a) r=5
(b) r^2 + z^2=100
(c) z=r

Answer 1

I'm thinking you're working in cylindrical coordinates. If you are, then r=5 means "r=5 no matter what theta or z is". If this is the case, you'd have a cylinder of radius 5, extending from -infinity to +infinity.

The second one you can look at by considering the definition of the transformation of coordinates from rectilinear to cylindrical, namely,\[x=r \cos \theta \]\[y= r \sin \theta \]\[z=z\]Then\[r^2+z^2=(x^2+y^2)+z^2=100\]This is just a sphere of radius 10.

The last one you can use the definitions again to find\[z=r \rightarrow z=\sqrt{x^2+y^2} \]which is a circular cone.

Answer 2

I only took the positive square root since you have to consider the fact that, with the definitions, r>0, so the representation of r in the Cartesian space must be positive also.

Answer 3

thank you soo much :)