Convert the equation p = 8 to rectangular coordinates and write in standard form.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
2.7.7
![**Problem Statement:**
Convert the equation \( \rho = 8 \) to rectangular coordinates and write in standard form.
**Solution:**
In spherical coordinates, \( \rho \) represents the radial distance from the origin to the point. The equation \( \rho = 8 \) describes a sphere centered at the origin with a radius of 8.
**Conversion to Rectangular Coordinates:**
To convert from spherical to rectangular coordinates, the formula is:
\[
x^2 + y^2 + z^2 = \rho^2
\]
Given:
\[
\rho = 8
\]
Substitute \( \rho \) in the formula:
\[
x^2 + y^2 + z^2 = 8^2
\]
Simplify:
\[
x^2 + y^2 + z^2 = 64
\]
**Conclusion:**
The equation of the sphere in rectangular coordinates is:
\[
x^2 + y^2 + z^2 = 64
\]
This represents a sphere with a center at the origin and a radius of 8 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F21e8a523-579f-40b9-8f0a-805524283a8d%2F58922b14-5159-44f6-8140-5b30a5ad6100%2F7rd838k_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Convert the equation \( \rho = 8 \) to rectangular coordinates and write in standard form.
**Solution:**
In spherical coordinates, \( \rho \) represents the radial distance from the origin to the point. The equation \( \rho = 8 \) describes a sphere centered at the origin with a radius of 8.
**Conversion to Rectangular Coordinates:**
To convert from spherical to rectangular coordinates, the formula is:
\[
x^2 + y^2 + z^2 = \rho^2
\]
Given:
\[
\rho = 8
\]
Substitute \( \rho \) in the formula:
\[
x^2 + y^2 + z^2 = 8^2
\]
Simplify:
\[
x^2 + y^2 + z^2 = 64
\]
**Conclusion:**
The equation of the sphere in rectangular coordinates is:
\[
x^2 + y^2 + z^2 = 64
\]
This represents a sphere with a center at the origin and a radius of 8 units.
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