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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Question
![**Problem Statement:**
Find a Cartesian equation for the curve.
\[ r = 9 \cos(\theta) \]
**Explanation:**
The provided equation is in polar form, where \( r \) represents the radial distance from the origin, and \( \theta \) is the angle from the positive x-axis. The goal is to convert this equation into Cartesian coordinates (x, y).
**Conversion Steps:**
1. Recall the polar to Cartesian coordinate transformations:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
- \( r^2 = x^2 + y^2 \)
2. From the given equation \( r = 9 \cos(\theta) \), multiply both sides by \( r \):
\[ r^2 = 9r \cos(\theta) \]
3. Substitute the Cartesian transformations:
- Replace \( r^2 \) with \( x^2 + y^2 \)
- Replace \( r \cos(\theta) \) with \( x \)
This yields:
\[ x^2 + y^2 = 9x \]
4. Rearrange the equation:
\[ x^2 - 9x + y^2 = 0 \]
This is the Cartesian equation for the given polar curve.
**Diagram:**
The diagram contains a blank rectangular box, which likely serves as a placeholder for graphing the polar or Cartesian equation. The graph of the equation in polar coordinates \( r = 9 \cos(\theta) \) typically represents a circle centered at \( (4.5, 0) \) with a radius of 4.5 in Cartesian coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf8dd5bb-1368-4a10-9929-9294802d0b74%2F26b6a1f9-8824-4e67-908b-affabe1313d2%2Fh6goeln_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a Cartesian equation for the curve.
\[ r = 9 \cos(\theta) \]
**Explanation:**
The provided equation is in polar form, where \( r \) represents the radial distance from the origin, and \( \theta \) is the angle from the positive x-axis. The goal is to convert this equation into Cartesian coordinates (x, y).
**Conversion Steps:**
1. Recall the polar to Cartesian coordinate transformations:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
- \( r^2 = x^2 + y^2 \)
2. From the given equation \( r = 9 \cos(\theta) \), multiply both sides by \( r \):
\[ r^2 = 9r \cos(\theta) \]
3. Substitute the Cartesian transformations:
- Replace \( r^2 \) with \( x^2 + y^2 \)
- Replace \( r \cos(\theta) \) with \( x \)
This yields:
\[ x^2 + y^2 = 9x \]
4. Rearrange the equation:
\[ x^2 - 9x + y^2 = 0 \]
This is the Cartesian equation for the given polar curve.
**Diagram:**
The diagram contains a blank rectangular box, which likely serves as a placeholder for graphing the polar or Cartesian equation. The graph of the equation in polar coordinates \( r = 9 \cos(\theta) \) typically represents a circle centered at \( (4.5, 0) \) with a radius of 4.5 in Cartesian coordinates.
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