For a constant \(c\) in spherical coordinates \( (\rho,\theta,\phi)\), find the shape described by the equation \(\phi = c\)
Because of the spherical coordinates, phi = c traces a cylinder. So the answer is (E) cylinder.
In spherical coordinates, the \(\phi\) represents the angle between (the line segment between the origin and a point) and the x-y plane.
If \(\phi = c\), that means the angle \(\phi\) is constant. Construct a line L such that the angle between L and the x-y plane is \(\phi\), then revolve it around the z-axis. For any line segment L0 in the resulting figure, the angle between L0 and x-y plane is \(\phi\). Therefore the resulting figure is a cone.
Therefore the answer is (F).