Let's analyze the two questions in the uploaded image step-by-step.

1. Surface $r = 4 \cos(\theta)$:

• This equation is given in polar coordinates.
• Type of surface: This describes a circle in the 2D $xy$-plane (more specifically, a shifted circle).
• To convert it to rectangular coordinates:
  1. Use the identities: $r^2 = x^2 + y^2, \quad x = r \cos(\theta)$
  2. Substituting $r = 4 \cos(\theta)$: $x = 4 \cos(\theta) \implies x^2 + y^2 = 4x$
  • Answer in rectangular coordinates:
    $x^2 + y^2 = 4x$
    (This equation describes a circle centered at $(2, 0)$ with radius 2.)

2. Surface $z = \sqrt{3x^2 + 3y^2}$:

• Type of surface: This describes a cone.

  • Since the $z$-value is proportional to the square root of $x^2 + y^2$, the shape forms a cone symmetric around the $z$-axis.

• Spherical coordinates:
  In spherical coordinates: $\tan \phi = \frac{\sqrt{x^2 + y^2}}{z}$ For this cone, the angle $\phi$ (the polar angle from the positive $z$-axis) is constant. We are given: $\phi = \frac{\pi}{6}$ This means the cone has an opening angle of $30^\circ$ from the vertical axis.

Would you like further clarification on any part?

Here are five questions you might find interesting:

1. How do polar and rectangular coordinates relate to each other in general?
2. How do you derive the equation of a circle from polar to rectangular coordinates?
3. What other surfaces can be expressed using cylindrical or spherical coordinates?
4. How does the angle $\phi$ affect the shape of a cone in spherical coordinates?
5. How would the equation of a cone change if it was oriented along a different axis?

Tip: Visualizing surfaces in multiple coordinate systems helps solidify your understanding of how geometry and algebra connect.