Determine Rectangular Equation from Parametrization2

Given the parametrization $x=3u\sin v$, $y=3u\cos v$, and $z=u^2$, determine the rectangular equation.

In this problem, you will explore the concept of converting a set of parametric equations into a single rectangular equation. Parametrization is a technique used frequently in calculus and 3-dimensional geometry to describe geometric forms such as curves and surfaces. In essence, a set of parametric equations expresses the coordinates of the points making up a geometric object as functions of one or more parameters. In this case, you're working with three parameters, describing a surface in three-dimensional space.

The challenge of this problem is to eliminate the parameters, which requires understanding the relationships between different components of the equations. One common strategy is to express one parameter in terms of another using trigonometric identities, and then substitute it back to find a single equation. Here, noticing the structure of the parametrization involving trigonometric functions of the same parameter, suggests this is forming a circle.

Additionally, it's useful to recognize that the parameterization in cylindrical coordinates often involves expressions of trigonometric terms. Transitioning from this perspective, you can utilize known identities, such as the Pythagorean identity, to aid in transforming and simplifying the expressions into Cartesian coordinates. The ultimate aim is to depict the surface using a relationship directly between x, y, and z without involving the parameters, creating a rectangular or implicit equation that gives insight into the geometry of the described object.

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