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Chapter 15: Problem 6

\(5-6\) Describe in words the surface whose equation is given. $$\rho=3$$

Short Answer

Expert verified
The surface is a sphere with radius 3 centered at the origin.

Step by step solution

01

Identify the Coordinate System

The given equation is \(\rho = 3\). This suggests that we are working in spherical coordinates, where \(\rho\) represents the radial distance from the origin.
02

Interpret the Equation in Spherical Coordinates

In spherical coordinates, \(\rho\) is the radial distance from the origin to a point in space. The equation \(\rho = 3\) implies that every point on the surface is precisely 3 units away from the origin.
03

Visualize the Surface

Since \(\rho\) is constant and equal to 3, this forms a sphere with radius 3, centered at the origin. The value of \(\rho\) determines the size of the sphere but not its position, as it's always centered at the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius
The concept of a radius is crucial in understanding spheres, especially when dealing with spherical coordinates. In simple terms, the radius of a sphere is the distance from its center to any point on its surface. This is a constant value throughout the surface of the sphere. In the given context, the radius is presented as a specific value, which is 3. This means that no matter where you are on the surface of this sphere, you will always be exactly 3 units away from the center, or the origin in mathematical terms.
  • The radius defines the size of the sphere. A larger radius means a bigger sphere.
  • A radius is always positive, as it is a measure of distance.
  • In spherical coordinates, the term for radius is often represented by the Greek letter \(\rho\), which is equivalent to the radial distance from the origin.
Sphere
A sphere is a fascinating geometric shape, perfectly symmetrical and round in all directions. It is defined as the set of all points in space that are equidistant from a common point, known as the center. In our exercise, the center of the sphere is at the origin, meaning it's the point (0, 0, 0) in a 3-dimensional space.
  • Imagine a hollow ball; every point on the surface is part of the sphere.
  • The equation \[\rho = 3\]indicates that the sphere has a radius of 3 and is centered at the origin.
  • All points on this sphere are the same distance (3 units) from the center.
Visualizing the sphere can help you understand how spherical coordinates work. It's all about radial symmetry, where everything is aligned around a central point.
Radial Distance
In spherical coordinates, radial distance, often denoted as \(\rho\), is a pivotal concept. It represents the straight-line distance from the center of the sphere to any point in space. This distance is crucial in defining the surface we are describing.
  • The value of \(\rho\) is constant for a sphere and equals the radius of the sphere.
  • Radial distance helps in positioning points in a 3D space, outlining the sphere's boundary.
  • For the sphere with the equation \[\rho = 3\]radial distance determines that each point on this surface is positioned exactly 3 units away from the center.
This uniform radial distance highlights the simplicity yet elegance of the spherical coordinate system, making it a preferred choice for describing spherical surfaces.

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