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Describe in words the surface whose equation is given. r = 6

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 given 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 $.

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