https://file.aiquickdraw.com/mathbot/file/1726235568456v7ih9kp8.jpg

To convert the equation \( x^2 + y^2 = 9 \) to spherical coordinates, we first need to express the variables \(x\) and \(y\) in terms of spherical coordinates using the following relations:

In spherical coordinates, the relations for \(x\), \(y\), and \(z\) are:

\[
x = \rho \sin \phi \cos \theta
\]
\[
y = \rho \sin \phi \sin \theta
\]
\[
z = \rho \cos \phi
\]

Given the equation \( x^2 + y^2 = 9 \), substitute \(x\) and \(y\) with their spherical coordinate equivalents:

\[
(\rho \sin \phi \cos \theta)^2 + (\rho \sin \phi \sin \theta)^2 = 9
\]

Simplifying the left-hand side:

\[
\rho^2 \sin^2 \phi (\cos^2 \theta + \sin^2 \theta) = 9
\]

Since \( \cos^2 \theta + \sin^2 \theta = 1 \), the equation becomes:

\[
\rho^2 \sin^2 \phi = 9
\]

Now, solving for \( \rho \):

\[
\rho^2 = \frac{9}{\sin^2 \phi}
\]

\[
\rho = \frac{3}{|\sin \phi|}
\]

This is the equation in spherical coordinates. It shows the relation between \( \rho \) and \( \phi \), and represents a cylindrical surface in spherical coordinates.

Would you like any further details on this? Here are some related questions to explore:

1. How do you convert the full 3D equation \( x^2 + y^2 + z^2 = 9 \) to spherical coordinates?
2. What is the significance of the variables \( \rho \), \( \phi \), and \( \theta \) in spherical coordinates?
3. How do you convert between spherical and cylindrical coordinate systems?
4. How do you interpret this equation geometrically in spherical coordinates?
5. Can you derive a similar equation for a cone or other shapes in spherical coordinates?

**Tip:** Remember that in spherical coordinates, \( \phi \) is the angle from the positive \( z \)-axis, while \( \theta \) is the angle in the \( xy \)-plane from the positive \( x \)-axis.

Convert the equation x^2 + y^2 = 9 to spherical coordinates and write in standard form, solving for ρ.

Learn how to convert the equation x^2 + y^2 = 9 into spherical coordinates, solving for ρ. This solution uses key concepts from algebra and spherical coordinate systems, covering trigonometric identities and equation transformations. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation