Describe the surface.
x2 + z2 = 6
sphere
ellipsoid    
hyperboloid
circular cylinder
elliptic cylinder
hyperbolic cylinder
parabolic cylinder
elliptic paraboloid
Correct: Your answer is correct.
Sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.)
(Write an equation for the cross section at 
y = −6
 using x and z.)		
  
Incorrect: Your answer is incorrect.
(Write an equation for the cross section at 
y = 0
 using x and z.)		
  
Incorrect: Your answer is incorrect.
(Write an equation for the cross section at 
y = 6
 using x and z.)

### 1. **Describe the Surface:**
   The given equation is:

   \[
   x^2 + z^2 = 6
   \]

   This is the equation of a **circular cylinder**. The cylinder has its axis along the \( y \)-axis, and it has a constant radius \(\sqrt{6}\) in the \( xz \)-plane.

### 2. **Sketch the Surface:**
   Since the equation represents a circular cylinder, the surface is made up of circles in the \( xz \)-plane for each fixed \( y \). The radius of each circle is \(\sqrt{6}\), and the center is at the origin in the \( xz \)-plane.

### 3. **Cross Sections at Specific Values of \( y \):**

   - **Cross Section at \( y = -6 \):**
     
     The equation of the cross section at \( y = -6 \) is the same as the original equation:
     \[
     x^2 + z^2 = 6
     \]
     This is a circle in the \( xz \)-plane with radius \(\sqrt{6}\) and centered at the origin.

   - **Cross Section at \( y = 0 \):**
     
     Similarly, the cross section at \( y = 0 \) is also:
     \[
     x^2 + z^2 = 6
     \]
     This is another circle with the same radius and center.

   - **Cross Section at \( y = 6 \):**
     
     The cross section at \( y = 6 \) is:
     \[
     x^2 + z^2 = 6
     \]
     Again, this is a circle with the same characteristics.

### Conclusion:
The surface described by the equation \( x^2 + z^2 = 6 \) is a **circular cylinder** with its axis along the \( y \)-axis. The cross sections at \( y = -6 \), \( y = 0 \), and \( y = 6 \) are all identical and represent circles in the \( xz \)-plane with a radius of \(\sqrt{6}\).

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### **Relative Questions:**
1. What would the equation of the surface look like if the cylinder were rotated about the \( x \)-axis instead?
2. How does the radius of the cylinder change if the equation becomes \( x^2 + z^2 = 9 \)?
3. What is the equation of a cylinder with its axis along the \( z \)-axis?
4. How would the equation change if the cylinder were elliptic instead of circular?
5. What are the intersections of this cylinder with the planes \( x = 0 \) and \( z = 0 \)?

### **Tip:**
When identifying the type of surface represented by an equation, it’s useful to isolate the variables that determine the shape in each plane, especially if it’s symmetric or follows a standard form like a cylinder, sphere, or paraboloid.

Describe the surface. x^2 + z^2 = 6 sphere ellipsoid hyperboloid circular cylinder elliptic cylinder hyperbolic cylinder parabolic cylinder elliptic paraboloid Correct: Your answer is correct. Sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) (Write an equation for the cross section at y = -6 using x and z.) Incorrect: Your answer is incorrect. (Write an equation for the cross section at y = 0 using x and z.) Incorrect: Your answer is incorrect. (Write an equation for the cross section at y = 6 using x and z.)

Explore the detailed solution for describing and sketching the surface defined by the equation x^2 + z^2 = 6, identifying it as a circular cylinder. This problem also covers the equations for cross-sections at specific y-values and is suitable for students in Grades 10-12.

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