<Question>
For a constant $c,$ in cylindrical coordinates $(r,\theta,z),$ find the shape described by the equation
\[z = c.\](A) Line
(B) Circle
(C) Plane
(D) Sphere
(E) Cylinder
(F) Cone

Enter the letter of the correct option.
</Question>

<think>
<Plan_1>
Okay, so I need to figure out what shape is described by the equation z = c in cylindrical coordinates. Let me start by recalling what cylindrical coordinates are. Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a third coordinate (usually denoted z) that represents the height above the polar plane.

<prompt_1.1> Recall the definitions of cylindrical coordinates and how they relate to Cartesian coordinates. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
In other words, a point in space is represented by (r, θ, z), where r is the radial distance from the z-axis, θ is the angular coordinate (like the angle in polar coordinates), and z is the height.

So, the equation given is z = c. That means, regardless of the values of r and θ, the z-coordinate is fixed at c. Hmm, so if I imagine moving around in space, but keeping z constant at c, what does that look like?

In Cartesian coordinates, if we have z = c, that's a plane parallel to the xy-plane. But here, we're in cylindrical coordinates. Let me think. Since r and θ can vary freely, but z is fixed, this should represent all points that are at a constant height z = c, regardless of their distance from the z-axis (r) and their angle θ. So, in three dimensions, that should be a plane parallel to the xy-plane but shifted up along the z-axis by c units.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, but let me make sure. Let me recall the conversion between cylindrical and Cartesian coordinates.

<prompt_2.1> Convert the cylindrical equation z = c to Cartesian coordinates to verify the shape. </prompt_2.1>

But hold on, let me check the answer choices. The options are Line, Circle, Plane, Sphere, Cylinder, Cone. So, according to my reasoning, it's a Plane, which is option C.

But wait, sometimes in these multiple-choice questions, there can be tricky options. Let me double-check.

<prompt_2.2> Re-evaluate the answer choices to ensure the correct answer is selected. </prompt_2.2>

But just to be thorough, let me consider each option:

<prompt_2.3> Consider each answer choice to ensure the correct answer is selected. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
The relationships are:

x = r cosθ

y = r sinθ

z = z

So, if z is fixed at c, then in Cartesian coordinates, the equation becomes z = c. That's indeed a plane. So, that's consistent with my earlier thought. So, in cylindrical coordinates, z = c is a plane parallel to the xy-plane at height c.
</execution_2.1>

<execution_2.2>
Is there a possibility that z = c could represent something else? For example, in spherical coordinates, z = c would be a sphere or a plane. But here, it's cylindrical coordinates. So, cylindrical coordinates are (r, θ, z). So, fixing z and letting r and θ vary is indeed a plane. 

Alternatively, maybe someone might confuse it with a cylinder. But a cylinder in cylindrical coordinates would be if r is fixed and z varies. For example, r = c would be a cylinder of radius c around the z-axis. But here, z is fixed, so it's a different scenario.
</execution_2.2>

<execution_2.3>
A) Line: A line in 3D space would require two coordinates to be fixed, but here only one is fixed (z = c). So, no, a line isn't right.

B) Circle: A circle in cylindrical coordinates would be if, say, r = c (fixed radius) and θ varies, but z is fixed. Wait, if r is fixed and θ varies, then in Cartesian terms, that would be a circle in a plane parallel to the xy-plane. However, in this case, z is fixed, so actually, if r and θ vary but z is fixed, then it's not a circle. If r is fixed and θ varies, then it's a circle. But here, r and θ can be anything, so it's not a circle.

C) Plane: As discussed, z = c is a plane. This seems correct.

D) Sphere: A sphere in Cartesian coordinates is defined by x² + y² + z² = r². In cylindrical coordinates, that would be r² + z² = R², so if we have z = c, that's not a sphere. A sphere would require r² + z² = R², so this is different. So D is incorrect.

E) Cylinder: A cylinder in cylindrical coordinates is usually represented by r = constant (a cylinder around the z-axis). For example, r = a is a cylinder of radius a. But here, z is fixed, not r. So if z is fixed, then for each fixed z = c, the set of points is all (r, θ, c), which is a plane. However, if we have both r and z fixed, that would be a circle (for fixed r and θ varying, but z fixed, no, wait, if both r and z are fixed, then θ can vary, which would be a circle in the plane z = c. But in this case, z is fixed, so the entire plane is z = c. So E is incorrect.

F) Cone: A cone would have a linear relationship between r and z, like r = kz or something.
But here z is fixed, so that's not a cone. So F is out.

Therefore, the correct answer is C) Plane.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2, execution_2.3:
Wait, but just to confirm, let me visualize.

<prompt_3.1> Visualize the shape described by the equation z = c in cylindrical coordinates. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Imagine moving along the z-axis; at each point z = c, you can have any r and θ. So, in three dimensions, that's a flat surface extending infinitely in all directions parallel to the xy-plane but at height z = c. Yes, that's a plane.
</execution_3.1>
</Execution_3>

<Final_answer>
Thus, the shape described by the equation $z = c$ in cylindrical coordinates is a plane, which corresponds to option $\boxed{\text{C}}$.
</Final_answer>
</think>