equation is given in spherical coordinates. Express
the equation in rectangular coordinates and sketch the graph. ■
27equation is given in spherical coordinates. Express
the equation in rectangular coordinates and sketch the graph. ρ = 3

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.

Learn how to convert a spherical coordinates equation to rectangular coordinates. Understand the relationship between spherical and rectangular coordinate systems. Includes a step-by-step solution and graph sketching for better visualization. Suitable for advanced high school students and beyond.

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