The equation given in spherical coordinates is:

$\rho = 3$

Step 1: Convert the Spherical Equation to Rectangular Coordinates

In spherical coordinates, $\rho$ represents the radial distance from the origin. The relationship between spherical coordinates $(\rho, \theta, \phi)$ and rectangular coordinates $(x, y, z)$ is given by the following equations:

$x = \rho \sin\phi \cos\theta$ $y = \rho \sin\phi \sin\theta$ $z = \rho \cos\phi$

Given $\rho = 3$, the equation simplifies to:

$\rho = 3$

This implies that:

$\sqrt{x^2 + y^2 + z^2} = 3$

Step 2: Simplify the Equation

Squaring both sides of the equation to eliminate the square root, we get:

$x^2 + y^2 + z^2 = 9$

Step 3: Identify the Shape and Sketch the Graph

The equation $x^2 + y^2 + z^2 = 9$ represents a sphere centered at the origin $(0,0,0)$ with a radius of 3 units.

Sketch of the Graph:

• The center of the sphere is at the origin $(0, 0, 0)$.
• The radius of the sphere is 3 units, meaning it extends 3 units in all directions from the center.
• The surface of the sphere is equidistant from the origin at every point.

Visual Representation

If you were to sketch this, you'd draw a sphere centered at the origin with a radius of 3 units. Every point on this sphere satisfies the equation $x^2 + y^2 + z^2 = 9$.

Would you like further details, or have any questions?

Follow-up Questions:

1. What are spherical coordinates, and how do they relate to rectangular coordinates?
2. How would the equation change if $\rho$ was different from 3?
3. What is the significance of the radius in the spherical coordinate system?
4. How would the graph change if the equation was $\rho = f(\theta, \phi)$?
5. What are some real-world applications of converting spherical to rectangular coordinates?

Tip: When converting between coordinate systems, always remember the geometric interpretation to understand the shapes described by the equations.