The image you've uploaded provides a set of equations and surface names, where you need to match the equations to their corresponding surfaces. Let’s go through the matches one by one:

1. $x^2 + y^2 + z^2 = R^2$ corresponds to sphere, as this is the equation for a sphere in 3D.
2. $x^2 + y^2 = R^2$ corresponds to circular cylinder, as this describes a cylinder where the radius in the x-y plane is constant.
3. $z = r$ corresponds to cone, as this represents a cone when expressed in cylindrical coordinates.
4. $r = R$ corresponds to cylinder in cylindrical coordinates.
5. $r = \sin(\theta)$ corresponds to a specific type of surface (likely sphere, but this is related to polar equations in 3D).

Now, there are two equations that do not match anything else:

• $\phi = \frac{\pi}{4}$ (likely refers to a constant angle in spherical coordinates).
• $\rho = R$ refers to a constant radius in spherical coordinates.

Let's look for the surfaces that go with these last two, as they don't have a direct match.

If you'd like further explanation or details on these, feel free to ask!

Relative Questions:

1. What is the general form of the equation of a cone?
2. How can you convert between cylindrical and spherical coordinates?
3. What surface is represented by the equation $\phi = \text{constant}$ in spherical coordinates?
4. What is the equation of a paraboloid, and how can we distinguish it?
5. How can $r = \sin(\theta)$ describe part of a sphere?

Tip:

Remember that equations in different coordinate systems (Cartesian, cylindrical, and spherical) often describe the same geometric surfaces!