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To convert the cylindrical equation \( r = 6 \) into rectangular coordinates, we use the relationship between cylindrical and rectangular coordinates:

1. In cylindrical coordinates:
   - \( r = \sqrt{x^2 + y^2} \)
   
Given that \( r = 6 \), we substitute this into the rectangular form:

\[
\sqrt{x^2 + y^2} = 6
\]

2. Squaring both sides to remove the square root:

\[
x^2 + y^2 = 36
\]

Thus, the equation in rectangular coordinates is:

\[
x^2 + y^2 = 36
\]

This represents a cylinder with radius 6 centered along the \( z \)-axis in three-dimensional space.

Would you like further details or have any questions about this conversion?

Here are some related questions that expand on this topic:

1. How do you convert other cylindrical coordinate equations to rectangular form?
2. What is the significance of the \( z \)-axis in cylindrical coordinates?
3. How would this equation differ if there were a \( z \)-term in the cylindrical equation?
4. What surfaces do equations of the form \( r = k \) (where \( k \) is a constant) represent?
5. How would you convert an equation in spherical coordinates to rectangular coordinates?

**Tip:** Remember that in cylindrical coordinates, \( r \) represents the radial distance from the \( z \)-axis, which is why equations like \( r = k \) create cylinders around this axis when converted to rectangular coordinates.

Find an equation in rectangular coordinates for the surface represented by the cylindrical equation r = 6.

Explore the process of converting the cylindrical equation r = 6 to rectangular coordinates. This solution explains how to rewrite the equation as x^2 + y^2 = 36, representing a cylinder with radius 6 centered along the z-axis in three-dimensional space. Suitable for high school students in Grades 11-12 studying coordinate transformations.

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