Which of the following are considered cylinders? A. 4𝑦2+𝑧2=4

B. 𝑥2+2𝑦2−3𝑧=−1

C. 4𝑥2+4𝑦2+2𝑧2=3

D. 𝑦=3𝑥2−2

E. 4𝑥2−𝑦2+4𝑧2=1

F. −(cos(−3𝑦))=−(𝑧+3)

G. None of the above.

To determine which of the given equations represent cylinders, we first need to review the general definition of a **cylinder** in mathematics.

A cylinder is a surface that extends infinitely in one direction (along a specific axis), and its cross-sections in planes perpendicular to that axis are curves (typically circles, ellipses, parabolas, or other conic sections).

### Now let's analyze each equation:

---

**A. \( 4y^2 + z^2 = 4 \)**

This equation describes a curve in the \( yz \)-plane, specifically an ellipse. Since \( x \) does not appear in the equation, this curve is extruded along the \( x \)-axis, forming a **cylinder**.  
**Conclusion: A is a cylinder.**

---

**B. \( x^2 + 2y^2 - 3z = -1 \)**

This equation involves all three variables \( x \), \( y \), and \( z \). It does not represent a surface where one variable is absent, which would be necessary for a cylindrical shape.  
**Conclusion: B is not a cylinder.**

---

**C. \( 4x^2 + 4y^2 + 2z^2 = 3 \)**

This equation involves all three variables \( x \), \( y \), and \( z \), and represents an ellipsoid, not a cylinder, as there is no "missing" variable to indicate extrusion along one axis.  
**Conclusion: C is not a cylinder.**

---

**D. \( y = 3x^2 - 2 \)**

This is a parabola in the \( xy \)-plane. Since \( z \) is missing, this parabola extends infinitely along the \( z \)-axis, forming a **parabolic cylinder**.  
**Conclusion: D is a cylinder.**

---

**E. \( 4x^2 - y^2 + 4z^2 = 1 \)**

This equation involves all three variables \( x \), \( y \), and \( z \). It represents a hyperboloid, not a cylinder, as all three variables are involved in a non-trivial way.  
**Conclusion: E is not a cylinder.**

---

**F. \( -( \cos(-3y) ) = -(z + 3) \)**

This equation can be simplified to \( \cos(3y) = z + 3 \), which represents a non-linear relationship between \( y \) and \( z \). Since all variables are involved and there is no clear extrusion along any axis, this is not a cylinder.  
**Conclusion: F is not a cylinder.**

---

### Final Answer:
The equations that represent cylinders are:  
**A. \( 4y^2 + z^2 = 4 \)**  
**D. \( y = 3x^2 - 2 \)**

Would you like more details or have any questions?

---

### Related Questions:
1. What are the different types of cylinders in mathematics?
2. How can you distinguish a hyperboloid from a cylinder?
3. What is the difference between an elliptic cylinder and a parabolic cylinder?
4. Can a cone be considered a cylinder in any special case?
5. How do cylinders appear in higher-dimensional spaces?

### Tip:
When identifying cylinders, look for an equation where one variable is missing, indicating that the shape extends infinitely along the axis of the missing variable.

Which of the following are considered cylinders? A. 4y^2 + z^2 = 4 B. x^2 + 2y^2 − 3z = −1 C. 4x^2 + 4y^2 + 2z^2 = 3 D. y = 3x^2 − 2 E. 4x^2 − y^2 + 4z^2 = 1 F. −(cos(−3y))=−(z+3) G. None of the above.

Determine which of the given equations represent cylinders by analyzing their structure and form. Learn how to identify cylindrical surfaces in algebra and geometry, focusing on equations like 4y^2 + z^2 = 4 and y = 3x^2 − 2. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation