Final answer:
To convert from rectangular to cylindrical coordinates, calculate the radial distance r and angular coordinate θ with r = √(x^2 + y^2) and θ = arctan(y/x). Express answers with the origin as the center. Cylinder properties involve calculating volume and surface area using these coordinates.
Step-by-step explanation:
To convert rectangular coordinates to cylindrical coordinates, we need to use the following relationships: r is the magnitude of the projection of the point onto the xy-plane, θ (theta) is the angular coordinate, and z remains the same as in the rectangular coordinates. Given the rectangular coordinates (x, y), we can calculate r using the Pythagorean theorem (r = √(x^2 + y^2)) and θ using the arctangent function (θ = arctan(y/x)), rounded to four decimal places. Expressing the answers in a cylindrical coordinate system with the origin at the center is simply realizing that the radial distance and angle are measured from this origin.
For a cylinder, consider the volume and surface area. The volume is the product of the base area and the height (V = πr^2h). The external surface area includes the top and bottom circles (2πr^2) and the side surface which is the perimeter of the base circle times the height (2πrh).