/*! 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 36 Complete the square to write the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 36

Complete the square to write the equation of the sphere in standard form. Find the center and radius. \(4 x^{2}+4 y^{2}+4 z^{2}-4 x-32 y+8 z+33=0\)

Short Answer

Expert verified
The center of the sphere is at point (0.5, 4, -1) and the radius is 22.

Step by step solution

01

Split and Arrange

First, separate and arrange the x, y, and z terms respectively:\n\n \(4x^{2} - 4x + 4y虏 - 32y + 4z^{2} + 8z + 33 = 0\) \n\nFactor out a 4 from each group: \(4(x^{2} - x) + 4(y^{2} - 8y) + 4(z^{2} + 2z) = -33\)
02

Completing the Square

To rewrite the equations in a form \((x - h)^{2}\), \((y - k)^{2}\), and \((z - l)^{2}\) we add and subtract specific values inside the parentheses. For x terms use \((\frac{1}{2})^{2}\)=0.25, for y terms use \((\frac{8}{2})^{2}\)=16, for z terms use \((\frac{2}{2})^{2}\)=1. Put these values back in the equation: \(4((x^{2} - x + 0.25 - 0.25) + 4(y^{2} - 8y + 16 - 16) + 4(z^{2} + 2z + 1 - 1)) = -33\) \n\nSimplifying, this becomes: \(4([(x - 0.5)^{2} - 0.25] + [(y - 4)^{2} - 16] + [(z + 1)^{2} - 1]) = -33\)
03

Rearrange the Equation Into Standard Form

We bring all the constants to the other side of the equation: \(4[(x - 0.5)^{2} + (y - 4)^{2} + (z + 1)^{2}] = 33 + 4*0.25 + 4*16 + 4*1\) \n\nSimplify the right side to: \((x - 0.5)^{2} + (y - 4)^{2} + (z + 1)^{2} = 22^{2}\)

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!

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 direction cosines of \(u\) and demonstrate that the sum of the squares of the direction cosines is 1. $$ \mathbf{u}=5 \mathbf{i}+3 \mathbf{j}-\mathbf{k} $$

In Exercises 75 and \(76,\) sketch the vector \(v\) and write its component form. \(\mathbf{v}\) lies in the \(y z\) -plane, has magnitude 2 , and makes an angle of \(30^{\circ}\) with the positive \(y\) -axis.

Think About It In Exercises \(65-68\), find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible. The solid between the spheres \(x^{2}+y^{2}+z^{2}=4\) and \(x^{2}+y^{2}+z^{2}=9,\) and inside the cone \(z^{2}=x^{2}+y^{2}\)

The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (2,-3,4),(0,1,2),(-1,2,0) $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal to \(\mathbf{w},\) then \(\mathbf{u}+\mathbf{v}\) is orthogonal to \(\mathbf{w}\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

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.