### Instruction: For a constant $c,$ in spherical coordinates $(\rho,\theta,\phi),$ find the shape described by the equation
\[\phi = c.\](A) Line
(B) Circle
(C) Plane
(D) Sphere
(E) Cylinder
(F) Cone

Enter the letter of the correct option.### Response: To solve this problem, I need to recall the definition of spherical coordinates and how they relate to Cartesian coordinates.
I know that $x = \rho \sin \phi \cos \theta, y = \rho \sin \phi \sin \theta, z = \rho \cos \phi.$
I also know that $\phi$ is the angle between the positive $z$-axis and the line joining the origin and the point, and that it ranges from $0$ to $\pi.$
So, if $\phi$ is constant, then that means the angle between the positive $z$-axis and the line joining the origin and the point is constant for all points on the shape.
This implies that the shape is a cone with the positive $z$-axis as its axis of symmetry, and with a half-angle of $c.$