Please answer the question below. Thank you! # Polar Equation of a Cartesian Curve## Problem StatementFind a polar equation for the curve represented by the given Cartesian equation.\[ x^2 + y^2 = 16 \]## ExplanationThe Cartesian equation given is:\[ x^2 + y^2 = 16 \]This equation represents a circle with a radius of 4 centered at the origin in the Cartesian coordinate system. ### Conversion to Polar CoordinatesTo convert this equation from Cartesian coordinates to polar coordinates, we use the relationship between Cartesian and polar forms:- $ x = r \cos(\theta) $- $ y = r \sin(\theta) $- $ r^2 = x^2 + y^2 $Substitute these into the given equation:\[ x^2 + y^2 = 16 \]\[ r^2 = 16 \]Therefore, the polar equation is:\[ r = 4 \]## Graphical RepresentationThe box below the question in the original image suggests there might be space for a graphical representation or a final answer box.In the polar coordinate system, $ r = 4 $ represents a circle with radius 4 centered at the origin (0, 0).### SummaryThe polar equation corresponding to the given Cartesian equation is:\[ \boxed{r = 4} \]This is an important concept in the conversion of coordinates and understanding the relationship between Cartesian and polar forms.

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Math

Calculus

Find a polar equation for the curve represented by the given Cartesian equation. x² + y2 = 16

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Please answer the question below. Thank you!

# Polar Equation of a Cartesian Curve

## Problem Statement

Find a polar equation for the curve represented by the given Cartesian equation.

\[ x^2 + y^2 = 16 \]

## Explanation

The Cartesian equation given is:

\[ x^2 + y^2 = 16 \]

This equation represents a circle with a radius of 4 centered at the origin in the Cartesian coordinate system.

### Conversion to Polar Coordinates

To convert this equation from Cartesian coordinates to polar coordinates, we use the relationship between Cartesian and polar forms:
- $ x = r \cos(\theta) $
- $ y = r \sin(\theta) $
- $ r^2 = x^2 + y^2 $

Substitute these into the given equation:

\[ x^2 + y^2 = 16 \]
\[ r^2 = 16 \]

Therefore, the polar equation is:

\[ r = 4 \]

## Graphical Representation

The box below the question in the original image suggests there might be space for a graphical representation or a final answer box.

In the polar coordinate system, $ r = 4 $ represents a circle with radius 4 centered at the origin (0, 0).

### Summary

The polar equation corresponding to the given Cartesian equation is:

\[ \boxed{r = 4} \]

This is an important concept in the conversion of coordinates and understanding the relationship between Cartesian and polar forms.

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