Here, \(2a = 3c \Rightarrow a = rac{3c}{2}\), so the shape is a sphere of radius \(rac{3c}{2}\) centered at \((0, 0, rac{3c}{2})\).

["Understanding the Geometry: How (2a = 3c \Rightarrow a = \frac{3c}{2}) Defines a Spherical Shape in 3D Space", "When dealing with geometric shapes in three-dimensional space, algebraic relationships often encode key spatial properties. Consider the equation (2a = 3c \Rightarrow a = \frac{3c}{2}). While seemingly simple, this relationship reveals important information about positioning and symmetry, particularly when interpreted in a coordinate system.", "### The Mathematical Interpretation", "The equation (2a = 3c) expresses a proportionality between the variables (a) and (c). Solving for (a), we obtain:\n[\na = \frac{3c}{2}\n]\nThis linear relationship indicates a fixed scaling between two geometric parameters. But when embedded into a spatial context—such as coordinates in 3D space—this ratio defines a specific geometric configuration: a sphere with radius ( \frac{3c}{2} ), centered at ((0, 0, \frac{3c}{2})).", "### Deriving the Sphere’s Geometry", "In a standard Cartesian coordinate system, a sphere is defined as the set of all points ((x, y, z)) that satisfy the equation:\n[\n(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\n]\nwhere ((h, k, l)) is the center and (r) is the radius.", "Let’s map the given center ((0, 0, \frac{3c}{2})) and substitute (r = \frac{3c}{2}):\n[\nx^2 + y^2 + \left(z - \frac{3c}{2}\right)^2 = \left(\frac{3c}{2}\right)^2\n]", "Now, recall from the original proportion that (a = \frac{3c}{2}). Thus, the radius simplifies to (a), and the full equation becomes:\n[\nx^2 + y^2 + z^2 - 3cz + \left(\frac{3c}{2}\right)^2 = \left(\frac{3c}{2}\right)^2\n]\nSubtracting (\left(\frac{3c}{2}\right)^2) from both sides yields:\n[\nx^2 + y^2 + z^2 - 3cz = 0\n]", "This confirms the standard form of a sphere centered along the (z)-axis. Importantly, by completing the square and aligning with the center at ((0, 0, \frac{3c}{2})), the equation describes a sphere with radius (\frac{3c}{2}).", "### Why This Center and Radius Matter", "- Center at ((0, 0, \frac{3c}{2})): This position places the sphere symmetrically along the positive (z)-axis, exactly halfway between the origin ((0, 0, 0)) and ((0, 0, 3c)). The height of the center above the (xy)-plane directly relates to (c), and the radius (\frac{3c}{2}) ensures the sphere touches key reference points in space.", "- Geometric Significance: The derived sphere passes through the origin and extends upward, spanning from (z = 0) (where it touches the (xy)-plane) to (z = 3c) (the topmost point), confirming consistency with the original parameter (a = \frac{3c}{2}) as the radius.", "### Practical Implications", "Such a geometric relationship is valuable in engineering, physics, and computer graphics, where predictable shapes simplify modeling and calculations. For instance:\n- In 3D modeling, aligning objects with spheres ensures balanced spatial distribution.\n- In physics, spherical symmetry often represents systems with uniform properties (like gravitational or electric fields).\n- Architectural designs may use such proportions to achieve aesthetic symmetry and structural stability.", "### Conclusion", "The simple equation (2a = 3c \Rightarrow a = \frac{3c}{2}), when interpreted in 3D space, defines a sphere of radius (\frac{3c}{2}) centered at ((0, 0, \frac{3c}{2})). This elegant connection between algebra and geometry illustrates how proportional relationships shape spatial form—transforming a formula into a tangible, symmetrical object. Understanding these principles empowers better visualization and application across disciplines where spatial reasoning is key."]