Problem 71 - Advanced Engineering Mathematics

Question

In Problems \(71-74\), convert the given equation to rectangular coordinates. $$ \rho=10 $$

Step-by-Step Solution

Answer

The equation \( \rho = 10 \) converts to \( x^2 + y^2 + z^2 = 100 \) in rectangular coordinates.

Step 1: Identify the Given Equation Type

The given equation is in spherical coordinates. In spherical coordinates, \( \rho \) represents the radial distance from the origin. Here, we have \( \rho = 10 \).

Step 2: Recall the Conversion Formulas

To convert from spherical to rectangular coordinates, we use the formulas: \( x = \rho \sin(\phi) \cos(\theta) \), \( y = \rho \sin(\phi) \sin(\theta) \), and \( z = \rho \cos(\phi) \).

Step 3: Substitute \( \rho = 10 \) in the Equations

Since \( \rho = 10 \), substitute it into each equation: \( x = 10 \sin(\phi) \cos(\theta) \), \( y = 10 \sin(\phi) \sin(\theta) \), and \( z = 10 \cos(\phi) \).

Step 4: Analyze the Resulting Surface

Notice that this represents a sphere in the rectangular coordinate system with radius 10 centered at the origin. The equation of a sphere in rectangular coordinates is \( x^2 + y^2 + z^2 = R^2 \). Here, \( R = 10 \), hence we have \( x^2 + y^2 + z^2 = 100 \).

Key Concepts

Spherical Coordinates

Spherical coordinates offer a different way to describe points in space, using three variables: \( \rho \), \( \phi \), and \( \theta \). Where:

• \( \rho \): The radial distance from the origin to the point.
• \( \phi \): The polar angle, measured from the positive \( z \)-axis.
• \( \theta \): The azimuthal angle, in the \( xy \)-plane, measured from the positive \( x \)-axis.

These coordinates are particularly useful when dealing with spherical objects or problems involving symmetry around a central point.
This is because they simplify the mathematics involved in analyzing spheres. In this exercise, you're given an equation \( \rho = 10 \), indicating all points at a fixed distance 10 from the origin. This outlines a perfect sphere in three-dimensional space.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in space using three variables \( x \), \( y \), and \( z \). These represent the distances along the respective axes:

• \( x \): Distance along the horizontal axis.
• \( y \): Distance along the vertical axis.
• \( z \): Distance along the depth axis.

These coordinates form the basis for most mathematical descriptions of space, where each point is uniquely defined.
In contrast to spherical coordinates, rectangular coordinates can be more straightforward for visualizing objects in space like boxes or planes.
Converting from spherical to rectangular coordinates can sometimes simplify the interpretation of a geometric shape.

Equation of a Sphere

An equation of a sphere in a rectangular coordinate system is typically represented as \( x^2 + y^2 + z^2 = R^2 \).
This equation signifies that every point on the sphere is equidistant from the center point, which in this case is the origin. Therefore:

• \( x^2 + y^2 + z^2 = R^2 \) means the radius is \( R \).
• Each point satisfying the equation lies on the surface of the sphere.

In the original exercise, with \( \rho = 10 \), the conversion results in \( x^2 + y^2 + z^2 = 100 \), indicating a sphere centered at the origin with a radius of 10 units.
This representation helps visualize the position and size of the sphere within the space.

Coordinate Conversion

Coordinate conversion involves translating a set of coordinates from one system to another, utilizing mathematical equations. For spherical to rectangular coordinates, the formulas are:

• \( x = \rho \sin(\phi) \cos(\theta) \)
• \( y = \rho \sin(\phi) \sin(\theta) \)
• \( z = \rho \cos(\phi) \)

These equations point to a direct relationship between the spherical and rectangular systems, allowing conversion between them. In our task, given a sphere of radius 10, substituting \( \rho = 10 \) into these equations translates spherical coordinates describing the sphere into its equivalent rectangular form.
Understanding how to convert between these systems enriches your ability to switch perspectives when analyzing geometric problems.