G write the equation in spherical coordinates. (a) 7z2 = 8x2 + 8y2

Answer:[tex]\displaystyle 7 \cos^2 \phi - 8 \sin^2 \phi = 0[/tex]General Formulas and Concepts:
Multivariable CalculusSpherical 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: DefineIdentify.[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.---Learn more about spherical coordinates: more about multivariable calculus: : Multivariable CalculusUnit: Triple Integrals Applications