Question

Find a spherical coordinate equation for the sphere x^(2) y^(2) (z-7)^(2)=49

Answer 1

The spherical coordinate equation for the sphere defined by x^2 + y^2 + (z-7)^2 = 49 is r = 7. This is derived by transforming the original Cartesian equation into spherical coordinates and recognizing that the radial distance r uniquely determines the size of the sphere.

Step-by-step explanation:

To find a spherical coordinate equation for the sphere given by the equation x2 + y2 + (z - 7)2 = 49, we must convert this equation from Cartesian coordinates to spherical coordinates. The relationship between Cartesian and spherical coordinates is expressed as x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, where r is the radial distance, θ is the polar angle, and φ is the azimuthal angle.

Upon substituting we find:

• r sin θ cos φ2 + r sin θ sin φ2 + (r cos θ - 7)2 = 49
• We simplify using the identity sin2(θ) + cos2(θ) = 1 and obtain r2 + (r cos θ - 7)2 = 49.
• The final spherical coordinate equation for the sphere is thus r = 7 since only the radial distance specifies the size of the sphere and the sphere is centered at the point (0, 0, 7) in Cartesian coordinates.