Answer:
[tex]\displaystyle 7 \cos^2 \phi - 8 \sin^2 \phi = 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
Identify.
[tex]\displaystyle 7z^2 = 8x^2 + 8y^2[/tex]
Step 2: Convert
- [Equation] Substitute in Spherical Coordinate Conversions:
[tex]\displaystyle 7( \rho \cos \phi )^2 = 8( \rho \sin \phi \cos \theta )^2 + 8( \rho \sin \phi \sin \theta )^2[/tex] - Simplify:
[tex]\displaystyle 7 \rho^2 \cos^2 \phi = 8 \rho^2 \sin^2 \phi \cos^2 \theta + 8 \rho^2 \sin ^2 \phi \sin^2 \theta[/tex] - Factor:
[tex]\displaystyle 7 \rho^2 \cos^2 \phi = 8 \rho^2 \sin^2 \phi \bigg( \sin^2 \theta + \cos ^2 \theta \bigg)[/tex] - Simplify:
[tex]\displaystyle 7 \rho^2 \cos^2 \phi = 8 \rho^2 \sin^2 \phi[/tex] - Simplify:
[tex]\displaystyle 7 \cos^2 \phi = 8 \sin^2 \phi[/tex] - Rewrite:
[tex]\displaystyle 7 \cos^2 \phi - 8 \sin^2 \phi = 0[/tex]
∴ we have found the given equation in terms of spherical coordinates.
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Learn more about spherical coordinates: https://brainly.com/question/9557773
Learn more about multivariable calculus: https://brainly.com/question/4746216
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Topic: Multivariable Calculus
Unit: Triple Integrals Applications
