Respuesta :

Space

Answer:

[tex]\displaystyle 4 \rho - 3 \sin \phi \cos \theta = 0[/tex]

General Formulas and Concepts:

Multivariable Calculus

Spherical Coordinate Conversions:

  • [tex]\displaystyle r = \rho \sin \phi[/tex]
  • [tex]\displaystyle x = \rho \sin \phi \cos \theta[/tex]
  • [tex]\displaystyle z = \rho \cos \phi[/tex]
  • [tex]\displaystyle y = \rho \sin \phi \sin \theta[/tex]
  • [tex]\displaystyle \rho = \sqrt{x^2 + y^2 + z^2}[/tex]

Step-by-step explanation:

Step 1: Define

[tex]\displaystyle 4x^2 - 3x + 4y^2 + 4z^2 = 0[/tex]

Step 2: Convert

  1. [Equation] Rewrite:
    [tex]\displaystyle 4x^2 + 4y^2 + 4z^2 - 3x = 0[/tex]
  2. [Equation] Factor:
    [tex]\displaystyle 4(x^2 + y^2 + z^2) - 3x = 0[/tex]
  3. [Equation] Substitute in Spherical Coordinate Conversions:
    [tex]\displaystyle 4 \rho^2 - 3 \rho \sin \phi \cos \theta = 0[/tex]
  4. Factor:
    [tex]\displaystyle \rho \big( 4 \rho - 3 \sin \phi \cos \theta \big) = 0[/tex]
  5. Simplify:
    [tex]\displaystyle 4 \rho - 3 \sin \phi \cos \theta = 0[/tex]

∴ we have converted the rectangular equation into spherical coordinates.

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Learn more about spherical coordinates: https://brainly.com/question/6093758

Learn more about multivariable calculus: https://brainly.com/question/4746216

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Topic: Multivariable Calculus

Unit: Triple Integrals Applications

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