Find an equation in rectangular coordinates for the cylindrical equation \( r=3 \)

In cylindrical coordinates, \( r = 3 \) represents all points that are at a distance of 3 units from the origin in the \( xy \)-plane, extending infinitely in the \( z \)-direction. To convert this to rectangular (Cartesian) coordinates, we use the relationship between cylindrical and rectangular coordinates: \[ x = r \cos \theta \] \[ y = r \sin \theta \] \[ z = z \] Given \( r = 3 \), we can express \( x \) and \( y \) in terms of \( r \): \[ x^2 + y^2 = r^2 \] Substituting \( r = 3 \): \[ x^2 + y^2 = 3^2 \] \[ x^2 + y^2 = 9 \] This equation represents a cylinder of radius 3 centered along the \( z \)-axis in rectangular coordinates. **Answer:** After simplifying, an equivalent rectangular equation is x² + y² = 9.