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Chapter 8: Problem 28

Describe in words the region of \(\mathbb{R}^{3}\) represented by the equations or inequalities. \(y^{2}+z^{2}=16\)

Short Answer

Expert verified
The region is an infinite cylinder along the \(x\)-axis with radius 4.

Step by step solution

01

Understand the Equation

The equation given is \(y^2 + z^2 = 16\). Recognize that this is the equation of a circle in a 2-dimensional \(yz\)-plane. The circle is centered at the origin (point \((0,0)\) on the \(yz\)-plane) and has a radius of \(4\), since \(4^2 = 16\).
02

Translate into 3D Space

Since the equation is given in \(\mathbb{R}^3\), consider how the circle in the \(yz\)-plane behaves in three dimensions. The parameter \(x\) is not explicitly stated in the equation, meaning it can take any value.
03

Describe the Region

The freedom of the \(x\)-value implies that for every \(x\), there is a circle of radius 4 in the \(yz\)-plane. This results in cylindrical symmetry around the \(x\)-axis. Therefore, the described region is a cylinder with infinite length along the \(x\)-axis and a circular cross-section of radius 4 centered at every point \(x\) on the \(yz\)-plane.

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Key Concepts

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

Cylindrical Symmetry in 3D Space
Cylindrical symmetry refers to the invariance of a shape or equation when rotated around a central axis. In our exercise, the axis is the x-axis. When a circle lies in the yz-plane and the circle's equation does not involve the x-coordinate, this circle effectively extends in both positive and negative x directions. This creates a cylindrical shape because at every point along the x-axis, the shape remains identical, hence forming a cylinder with no beginning or end along the x-axis.

Key features of cylindrical symmetry in 3D are:
  • All cross-sections perpendicular to the axis of symmetry are identical circles.
  • The object is invariant under rotations around the axis.
Understanding this concept helps us see how simpler 2D shapes can extend into 3D regions under certain symmetrical conditions.
Equation of a Circle in the yz-plane
The equation \(y^2 + z^2 = 16\) defines a circle in the yz-plane. This is a fundamental concept in geometry, where equations define shapes. In this case, we have a circle with the following properties:
  • Center at the origin, point (0,0) of the yz-plane.
  • Radius calculated as 4, because \(4^2 = 16\).
It's crucial to recognize the standard form of a circle's equation, \(x^2 + y^2 = r^2\), where r is the radius. This foundational understanding provides clarity on how 2D shapes translate into more complex 3D forms, like cylindrical symmetry along an axis.

Whenever the x-coordinate is missing from an equation in 3D, it indicates that the circle is free to extend along the x-axis, forming a cylinder.
Infinite Cylinder Along the x-axis
In the context of the given equation \(y^2 + z^2 = 16\), we derive an infinite cylinder. This concept describes a cylindrical shape that infinitely extends along the x-axis in both directions.

The defining characteristics of an infinite cylinder include:
  • The base circle is of radius 4, located in the yz-plane.
  • The cylinder extends indefinitely along the x-axis.
This means that for any x-value, the cylinder would look the same in perpendicular yz-planes. Such visualization is crucial in understanding how simple equations not containing one of the variables expand into three dimensions, creating infinite geometric regions.
Planes in Three Dimensions and Their Role
When dealing with three-dimensional space, planes play a crucial role in defining shapes and intersections. A plane is, essentially, a flat, two-dimensional slice through three-dimensional space. In our context, the yz-plane is where the circle equation \(y^2 + z^2 = 16\) resides.

Key points about planes in 3D:
  • Any plane can be visualized as an endless sheet stretching through space.
  • Planes can intersect with other planes, causing a line, or with other shapes, creating various intersections.
Understanding planes helps unravel how a 2D circle described in a plane can evolve into 3D cylindrical formations by extending infinitely along an unaccounted axis.

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