/*! 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 18 Find the equation of the sphere ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 18

Find the equation of the sphere with radius 2 and centered at (1,0,0)

Short Answer

Expert verified
The equation of the sphere is \((x - 1)^2 + y^2 + z^2 = 4\).

Step by step solution

01

Identify the General Equation of a Sphere

The general equation of a sphere with center \((h, k, l)\) and radius \(r\) is given by:\[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]
02

Substitute the Center Coordinates

Given that the center of the sphere is at \((1, 0, 0)\), substitute \(h = 1\), \(k = 0\), and \(l = 0\) into the general equation. This gives:\[(x - 1)^2 + (y - 0)^2 + (z - 0)^2 = r^2\]
03

Substitute the Radius

The radius of the sphere is given as \(2\). Substitute \(r = 2\) into the equation:\[(x - 1)^2 + y^2 + z^2 = 2^2\]
04

Simplify the Equation

Square the radius to simplify the equation:\[(x - 1)^2 + y^2 + z^2 = 4\]

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.

Center of the Sphere
Understanding the center of a sphere is fundamental when describing its equation. The center is simply a point in three-dimensional space. It's typically denoted as \((h, k, l)\). This is crucial because it indicates where the sphere is positioned in space.
For example, in our exercise, the center of the sphere is given as \(1, 0, 0\). It means that:
  • The sphere's center lies 1 unit along the x-axis.
  • It remains on the y-axis and z-axis at 0.

Knowing the center allows us to shift the general equation of a sphere to this specific location. It essentially determines where the entire structure of the sphere will be anchored in the coordinate system.
Radius of the Sphere
The radius of a sphere is the distance from its center to any point on its surface. Represented by \(r\), this value is integral when it comes to the equation of the sphere. In simplest terms, the radius stretches out equally in all directions from the center.
In our given scenario, the radius is \(2\). This implies:
  • The sphere extends 2 units from the center in every direction.
  • This uniform distance ensures the spherical shape is maintained.
The radius is squared in the general equation. This is an essential step since distances are typically squared in distance formulas relevant to geometry to eliminate any negative values, ensuring we work with positive lengths.
General Equation for a Sphere
To describe any sphere mathematically, we use a bespoke formula known as the general equation of a sphere. This formula integrates both the location of the sphere鈥檚 center \((h, k, l)\) and its radius \(r\).
The equation is: \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]This structure helps in deriving a specific sphere's equation byidentifying its center and radius. Let's break it down further:
  • The terms \(x - h\), \(y - k\), and \(z - l\) shift the basic sphere centered at the origin to its actual center.
  • The right side of the equation, \(r^2\), represents the square of the radius, which ensures all points \(x, y, z\) satisfying this equation are precisely at a distance \(r\) from the center.
In our exercise, substituting the sphere's center \(1, 0, 0\) and radius \(2\) results in:\[(x - 1)^2 + y^2 + z^2 = 4\]This expresses the specific dimensions and location of the sphere within our three-dimensional space.

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

Does the line parallel to the \(y\) -axis through the point (2,1,4) intersect the plane \(y=5 ?\) If so, where?

What do the level surfaces of \(f(x, y, z)=x^{2}-y^{2}+z^{2}\) look like? [Hint: Use cross-sections with \(y\) constant instead of cross-sections with \(z \text { constant. }]\)

Explain what is wrong with the statement. The level surfaces of \(f(x, y, z)=x^{2}+y^{2}\) are paraboloids.

Explain what is wrong with the statement. The graph of a function \(f(x, y, z)\) is a surface in 3 space.

Determine whether there is a value for the constant \(c\) making the function continuous everywhere. If so, find it. If not, explain why not. $$f(x, y)=\left\\{\begin{array}{ll}c+y, & x \leq 3 \\ 5-x, & x>3\end{array}\right.$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.