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Cartesian coordinates are a fundamental system used to describe the location of a point in three-dimensional space using three variables:

• \(x\) - the horizontal position
• \(y\) - the vertical position
• \(z\) - the depth position

In the case of the given equation \(x^2 + y^2 + z^2 = 144\), we are working with a system that measures distances from a single origin point, situated at (0, 0, 0). Each coordinate describes a perpendicular distance along one of the three axes that define the space.

The equation \(x^2 + y^2 + z^2 = 144\) represents a sphere. Here, it is crucial to recognize that in Cartesian coordinates, the expression \(x^2 + y^2 + z^2 = R^2\) describes a sphere centered at the origin with a radius given as \(R\). In this particular problem, the sphere's radius is 12, as 12 squared equals 144.

Converting between different coordinate systems allows us to better understand and solve problems within various contexts.

**Converting to Cylindrical Coordinates**: Cylindrical coordinates provide a natural extension of two-dimensional polar coordinates to three dimensions, using a radius \(r\), an angle \(\theta\), and a height \(z\).

• The relationship between Cartesian and Cylindrical coordinates is defined by:
  • \(x = r \cos(\theta)\)
  • \(y = r \sin(\theta)\)
  • \(z = z\)

For the sphere equation, after replacing \(x\) and \(y\) with the corresponding expressions, we use the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\), simplifying the expression to \(r^2 + z^2 = 144\).

**Converting to Spherical Coordinates**: Spherical coordinates use three parameters: the radius \(\rho\), the inclination \(\phi\), and the azimuthal angle \(\theta\).

• The conversion formulas from Cartesian to Spherical coordinates are:
  • \(x = \rho \sin(\phi) \cos(\theta)\)
  • \(y = \rho \sin(\phi) \sin(\theta)\)
  • \(z = \rho \cos(\phi)\)

By plugging into the sphere equation, \(x^2 + y^2 + z^2\) becomes \(\rho^2\), which simplifies directly to \(\rho = 12\), when equated to 144.

Understanding the spherical equation in different coordinate systems unveils how geometry describes three-dimensional shapes.

The general equation for a sphere in Cartesian coordinates, when centered at the origin, is \(x^2 + y^2 + z^2 = R^2\). This represents all points that are a fixed distance \(R\) from the origin. In our example, with \(R = 12\), this describes a sphere with a radius of 12 units.

When we move to cylindrical coordinates, the equation gains simplicity: \(r^2 + z^2 = R^2\). Here, \(r\) represents the radial distance from the \(z\)-axis, and \(z\) represents the height. This illustrates how the cylinder wraps around the \(z\) axis, with \(r\) and \(z\) forming right angles to complete the hemisphere in 3D space.

The spherical coordinate conversion reveals the most apparent form of a sphere: \(\rho = R\). \(\rho\) stands as the direct distance from the origin to any point on the sphere's surface, confirming \(\rho = 12\). Using spherical coordinates, the notion of radii becomes clearer, emphasizing symmetry from the origin point.