The aim of this question is to **infer/visualize the shapes/surfaces** constructed from a given mathematical function using prior knowledge of standard functions.

The standard equation of a **circle in two-dimensional plane** is given by:

\[ x^2 \ + \ y^2 \ = \ r^2 \ … \ … \ … \ ( 1 )\]

The standard equation of a **sphere in three-dimensional space** is given by:

\[ x^2 \ + \ y^2 \ + \ z^2 = \ r^2 \ … \ … \ … \ ( 2 )\]

We will use both of these equations for solving the given question.

## Expert Answer

**Given:**

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

**Substituting $ r \ = \ 6 $:**

\[ x^2 \ + \ y^2 \ = \ ( 6 )^2 \]

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

**Part (a)**: Describing the given equation in a **two-dimensional plane.**

**Compared with equation no. (1)**, we can see that the **g****iven equation represents a circle** located at the origin with a radius of 6.

**Part (b)**: Describing the given equation in a **three-dimensional space.**

**Compared with equation no. (2)**, we can see that the **given equation is not a sphere** since the third axis $ z $ is missing.

Using information **from part (a),** we can see that the **given equation represents a circle located in the xy-plane** with a radius of 6 for a given fixed value of $ z $.

Since $ z $ can vary from $ – \infty $ to $ + \infty $, we can **stack such circles along z-axis**.

Hence, we can conclude that the **given equation represents a cylinder** with radius $ 6 $ extending from from $ – \infty $ to $ + \infty $ along $ z-axis $.

## Numerical Result

The **given equation represents a cylinder** with radius $ 6 $ extending from $ – \infty $ to $ + \infty $ along $ z-axis $.

## Example

Describe the following equation in words (assume $ r \ = \ 1 $ ):

\[ \boldsymbol{ x^2 \ + \ z^2 \ = \ r^2 } \]

**Substituting $ r \ = \ 1 $:**

\[ x^2 \ + \ z^2 \ = \ ( 1 )^2 \]

\[ \Rightarrow x^2 \ + \ z^2 \ = \ 1 \]

Compared with equation (1), we can see that the **given equation represents a circle located in the xz-plane** with a radius of 1 for a given fixed value of $ y $.

Since, $ y $ can vary from $ – \infty $ to $ + \infty $, we can **stack such circles along y-axis**.

Hence, we can conclude that the **given equation represents a cylinder** with radius $ 6 $ extending from from $ – \infty $ to $ + \infty $ along $ y-axis $.