Math Problem Statement

Describe the surface given by the equation x^2 + z^2 = 9 and identify the correct sketch.

Solution

The equation given is:

x2+z2=9x^2 + z^2 = 9

This is the equation of a circular cylinder. It represents a cylinder with radius 3 (since 9=329 = 3^2) oriented along the y-axis (as the variable yy is absent from the equation, indicating no variation in that direction).

Correct options:

  1. Surface Description: The correct option is D. circular cylinder.

  2. Sketch the Surface: Among the four 3D plots shown (A, B, C, and D), D is the plot of a circular cylinder, as it depicts a surface that wraps around an axis (likely the y-axis), matching the given equation x2+z2=9x^2 + z^2 = 9.

Would you like further explanation or details about cylindrical surfaces?


Here are 5 follow-up questions to expand on this:

  1. How does the absence of the yy variable affect the shape of the surface?
  2. What is the general form of a cylinder equation in 3D space?
  3. Can the equation x2+z2=r2x^2 + z^2 = r^2 represent any other geometric shape in different contexts?
  4. How would the graph change if the equation included y2y^2?
  5. What are the distinguishing features of an elliptic cylinder?

Tip: When identifying 3D surfaces, always check for missing variables, as they often indicate an extrusion along the corresponding axis!

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Cylinders
3D Surfaces

Formulas

x^2 + z^2 = r^2 (circular cylinder)

Theorems

Understanding of geometric surfaces defined by equations

Suitable Grade Level

Undergraduate