Find an equation in rectangular coordinates for the cylindrical equation \( r=4 z \)

To convert the cylindrical equation \( r = 4z \) into rectangular coordinates, we can use the relationships \( r = \sqrt{x^2 + y^2} \) and \( z = z \). Substituting these into the equation gives us \( \sqrt{x^2 + y^2} = 4z \). To eliminate the square root, we can square both sides, resulting in \( x^2 + y^2 = 16z^2 \). Therefore, the rectangular equation is \( x^2 + y^2 = 16z^2 \). An interesting aspect of this equation is that it represents a double cone with the vertex at the origin. As z increases, the radius of the circular cross-section increases, giving the structure a conical shape extending both above and below the xy-plane. You can imagine slicing through this conical shape at different heights and observing circular rotations. In practice, equations like \( x^2 + y^2 = 16z^2 \) have applications in 3D modeling and computer graphics, where understanding the geometry of shapes is crucial. This equation helps in creating 3D objects, like cones in animations or virtual environments, providing a basis for rendering realistic shapes based on mathematical equations!