/*! 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 50 Sketch the \(x y\) -trace of the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 50

Sketch the \(x y\) -trace of the sphere. $$ x^{2}+y^{2}+z^{2}-4 y+2 z-60=0 $$

Short Answer

Expert verified
The \(xy\)-trace of the sphere is a circle with a radius of 8 and its center at the point (0,2).

Step by step solution

01

Set z to zero

The equation of the sphere was given as \(x^{2}+y^{2}+z^{2}-4 y+2 z-60=0\). In order to find the \(xy\)-trace, set \(z=0\). This gives the new equation: \(x^{2}+y^{2}-4 y-60=0\)
02

Continue to complete the square

Completing the square gives a clearer view of the properties of the circle. Rewrite the equation as follows: \(x^{2}+(y-2)^{2}-64=0\)
03

Write the equation in standard form

The equation \(x^{2}+(y-2)^{2}-64=0\) can be rewritten into the standard form of a circle \(x^{2}+(y-\delta)^{2}=\rho^{2}\), where \(\delta\) is the y-value of the center of the circle and \(\rho\) is the radius. This gives \((x-0)^{2}+(y-2)^{2}=8^{2}\), which has the center at the point (0,2) and the radius as 8.
04

Sketch the circle

Now draw a circle on the \(xy\)-plane. The center of the circle must be at (0,2) and the radius of the circle will be 8 units. This will be the \(xy\)-trace of the sphere.

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

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (-3,4),(-1,2),(1,1),(3,0) $$

Use the regression capabilities of a graphing utility or a spreadsheet to find linear and quadratic models for the data. State which model best fits the data. $$ (0,769),(1,677),(2,601),(3,543),(4,489),(5,411) $$

The revenues \(y\) (in millions of dollars) for Earthlink from 2000 through 2006 are shown in the table. $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 \\ \hline \text { Revenue, } y & 986.6 & 1244.9 & 1357.4 & 1401.9 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \text { Year } & 2004 & 2005 & 2006 \\ \hline \text { Revenue, } y & 1382.2 & 1290.1 & 1301.3 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility or a spreadsheet to create a scatter plot of the data. Let \(t=0\) represent the year 2000 . (b) Use the regression capabilities of a graphing utility or a spreadsheet to find an appropriate model for the data. (c) Explain why you chose the type of model that you created in part (b).

Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (1,36),(2,10),(3,0),(4,4),(5,16),(6,36) $$

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int \frac{y}{1+x^{2}} d A\\\ &R: \text { region bounded by } y=0, y=\sqrt{x}, x=4 \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics 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.