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Chapter 12: Problem 32

\(23-32\) Describe in words the region of \(\mathbb{R}^{3}\) represented by the equation or inequality. $$x^{2}+y^{2}+z^{2}>2 z$$

Short Answer

Expert verified
The region is outside a sphere of radius 1 centered at (0, 0, 1) in \\( \mathbb{R}^{3} \\\).

Step by step solution

01

Rewrite the expression

Start by rewriting the given inequality. The expression provided is \( x^2 + y^2 + z^2 > 2z \). You can rewrite this by subtracting \(2z\) from both sides: \( x^2 + y^2 + z^2 - 2z > 0 \).
02

Identify the geometric shape

Look for a common geometric form in the inequality. Recognize that \(x^2 + y^2 + (z-1)^2\) is a shifted sphere equation: \( x^2 + y^2 + (z-1)^2 = 1 \). This represents a sphere with radius 1 centered at \((0, 0, 1)\).
03

Describe the region

The inequality \( x^2 + y^2 + z^2 - 2z > 0 \) is modified to \( x^2 + y^2 + (z-1)^2 > 1 \). This means we look at points outside the sphere of radius 1 centered at \((0, 0, 1)\) because inequality indicates all points greater than this given sphere.

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Key Concepts

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

sphere equation
The sphere equation in three-dimensional space is a fundamental mathematical formula used to define the surface of a sphere. A sphere is the set of all points in 3D space that are equidistant from a fixed point, known as the center. This distance is called the radius. The standard equation of a sphere with center
  • at the point \( (h, k, l) \)
  • and radius \( r \)
can be expressed as: \[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]In the original exercise, the rewritten equation \( x^2 + y^2 + (z-1)^2 = 1 \) fits this formula, indicating a sphere with:
  • center located at \( (0, 0, 1) \)
  • and radius 1.
Such spheres form integral parts of geometry, providing a basic understanding of spatial relationships by defining boundaries and regions.
coordinate system
The coordinate system is a way to uniquely identify points in space using a set of numbers. In three-dimensional space, a
  • coordinate system consists of three axes: the x-axis, y-axis, and z-axis.
  • Each point in this space is determined by a triplet \( (x, y, z) \).
These coordinates represent the position of a point either
  • above or below the xy-plane along the z-axis,
  • left or right of the yz-plane along the x-axis,
  • and forward or backward along the y-axis.
The notion of a coordinate system is critical when describing geometric shapes like spheres or determining spatial regions. By providing a concrete numerical representation of space, it allows us to apply mathematical techniques to visualize and solve spatial problems more effectively.
3D inequalities
3D inequalities describe regions in three-dimensional space and are often used to define areas outside or inside geometric shapes like spheres or cubes. An inequality in 3D can show how a certain point relates to a boundary.In the given exercise, the inequality \( x^2 + y^2 + z^2 > 2z \) was rewritten to \( x^2 + y^2 + (z-1)^2 > 1 \).This inequality tells us about the points lying
  • outside the sphere centered at \( (0, 0, 1) \) with a radius of 1.
  • In 3D inequalities, when you see a symbol like \( > \), it indicates locations that are outside a defined boundary.
Thus, in this context, the inequality helps differentiate the space that lies beyond the surface of the sphere, visually helping understand the three-dimensional layout of regions.

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Most popular questions from this chapter

\(29-36\) Reduce the equation to one of the standard forms, classify the surface, and sketch it. $$4 x^{2}+y^{2}+4 z^{2}-4 y-24 z+36=0$$

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