/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 In Exercises \(29-32,\) find the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 29

In Exercises \(29-32,\) find the standard equation of the sphere. Center: (0,2,5) Radius: 2

Short Answer

Expert verified
The standard equation of the sphere is \(x^2 + (y - 2)^2 + (z - 5)^2 = 4

Step by step solution

01

Identify values of the parameters

Identify the given values. The center coordinates (h, k, l) are (0, 2, 5) correspondingly and the radius r is 2.
02

Substitution

Substitute the given values into the standard equation of a sphere. The formula is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). By substituting, it will be \((x - 0)^2 + (y - 2)^2 + (z - 5)^2 = 2^2\)
03

Simplify the equation

Simplify the euation, so we get \(x^2 + (y - 2)^2 + (z - 5)^2 = 4.\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding the Standard Form Equation of a Sphere
The standard form equation of a sphere is crucial in understanding and describing spherical shapes in three-dimensional space. This equation is expressed as:
  • \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\)
In this equation:
  • \( (h, k, l) \) represents the center of the sphere in three-dimensional space.
  • \( r \) is the radius, which is the distance from the center to any point on the sphere.
The equation ensures that every point \( (x, y, z) \) on the sphere is equidistant from the center, precisely \( r \) units away. In the problem provided, substituting the given center \( (0, 2, 5) \) and radius \( r = 2 \) into this equation gives:
  • \((x - 0)^2 + (y - 2)^2 + (z - 5)^2 = 2^2\)
Simplified, it becomes \(x^2 + (y - 2)^2 + (z - 5)^2 = 4\). This equation effectively describes a sphere centered at \( (0, 2, 5) \) with a radius of 2 in a three-dimensional space.
Exploring Three-Dimensional Geometry
Three-dimensional geometry allows us to explore shapes and forms that have depth, breadth, and height. It is a step beyond the two-dimensional space and introduces a deeper level of complexity.
  • In three-dimensional space, objects like spheres, cubes, and pyramids have volume, which is a measure of the space enclosed within the surface of these objects.
  • The coordinate system in this geometry uses three axes - typically labeled as x, y, and z - to plot any point in the space.
A sphere, in this context, is like an inflated ball, with every point on its surface being exactly the same distance from its center.
  • Adding the third dimension (z-axis) allows us to define points not just in left-right or forward-backward directions, but also up and down.
  • The introduction of depth (or height when considering the y-axis as vertical) makes these figures more reflective of real-world objects.
Interpreting spheres in three-dimensional geometry involves both spatial reasoning and understanding the symmetry that results from their standard form equations.
Navigating the Coordinate System
A coordinate system is essential for placing and describing objects in space. In a three-dimensional coordinate system, every point is defined by a set of three numbers, referred to as coordinates. These coordinates correspond to:
  • The x-coordinate, which indicates the horizontal position.
  • The y-coordinate, which represents the vertical position, often thought of as depth or height.
  • The z-coordinate, which adds the third dimension of forward or backward position.
By using these coordinates, the point \( (0, 2, 5) \) represents a precise position in three-dimensional space, marking the center of the sphere given in the exercise problem.
  • The inclusion of all three axes allows for a comprehensive system that can accurately describe any point in space, regardless of its position relative to the origin \((0, 0, 0)\).
      • Each managed axis helps us to describe positions and distances in a way that can be easily visualized and understood. This understanding is abstract in calculations but grounded and highly practical in fields ranging from engineering to computer graphics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(\mathbf{z}=5 \mathbf{u}-3 \mathbf{v}-\frac{1}{2} \mathbf{w}\)

Find \((\mathbf{a}) \mathbf{u} \cdot \mathbf{v},(\mathbf{b}) \mathbf{u} \cdot \mathbf{u},(\mathbf{c})\|\mathbf{u}\|^{2},(\mathbf{d})(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\) and \((e) u \cdot(2 v)\). $$ \mathbf{u}=\langle-4,8\rangle, \quad \mathbf{v}=\langle 6,3\rangle $$

Find the angle \(\theta\) between the vectors. $$ \begin{array}{l} \mathbf{u}=\langle 1,1,1\rangle \\ \mathbf{v}=\langle 2,1,-1\rangle \end{array} $$

Write an equation whose graph consists of the set of points \(P(x, y, z)\) that are twice as far from \(A(0,-1,1)\) as from \(B(1,2,0)\)

Find \((\mathbf{a}) \mathbf{u} \cdot \mathbf{v},(\mathbf{b}) \mathbf{u} \cdot \mathbf{u},(\mathbf{c})\|\mathbf{u}\|^{2},(\mathbf{d})(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\) and \((e) u \cdot(2 v)\). $$ \begin{array}{l} \mathbf{u}=2 \mathbf{i}+\mathbf{j}-2 \mathbf{k} \\ \mathbf{v}=\mathbf{i}-3 \mathbf{j}+2 \mathbf{k} \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.