https://file.aiquickdraw.com/mathbot/file/1729533541501m4mkjtel.jpg

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.)

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### 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.

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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.

Identify the surface in R^3: 1. The surface r = 4cos(θ) is a circle and the equation in rectangular coordinates is x^2 + y^2 = 4x. 2. The surface z = sqrt(3x^2 + 3y^2) is a cone and the equation in spherical coordinates is φ = π/6.

Learn how to identify surfaces in R^3, including a circle and a cone, and convert their equations between polar, rectangular, and spherical coordinates. This problem is suited for students studying Multivariable Calculus or Calculus III.

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