/*! 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 81 Identify and briefly describe th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 81

Identify and briefly describe the surfaces defined by the following equations. $$y=4 z^{2}-x^{2}$$

Short Answer

Expert verified
The surface described by the equation $$y = 4z^2 - x^2$$ is an elliptical paraboloid.

Step by step solution

01

Rewrite the given equation

The equation for the surface is given as: $$y = 4z^2 - x^2$$
02

Compare with standard equations

Compare the given equation with some well-known standard equations to identify the surface type. In this case, the equation appears to be similar to the equation of a paraboloid. To get a better understanding, we will rewrite the equation in the form of a standard paraboloid equation. The standard equation of an elliptic paraboloid is given by: $$\frac{x^2}{a^2} - \frac{y}{c} + \frac{z^2}{b^2} = 0$$
03

Rewrite the given equation in standard paraboloid form

To change the given equation to the standard paraboloid form, follow these steps: 1. Move \(x^2\) and \(4z^2\) to the right side of the equation. 2. Divide and adjust the equation to match the standard form. This gives us the equation: $$y = -x^2 + 4z^2$$ $$\Rightarrow \frac{-x^2}{1} + 4\frac{z^2}{1} = -y$$ $$\Rightarrow \frac{x^2}{1} - \frac{y}{4} + \frac{z^2}{\frac{1}{4}} = 0$$ Now, comparing with the standard equation, we see that: $$a^2 = 1, b^2 = \frac{1}{4}, c = 4$$
04

Describe the surface

Based on the standard paraboloid equation and the derived values for \(a, b,\) and \(c\), we can now describe the surface. We have an elliptical paraboloid centered at the origin (0, 0, 0) with its axis parallel to the y-axis, opening in the negative y-direction. The surface defined by the equation $$y = 4z^2 - x^2$$ is an elliptical paraboloid.

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

Let \(E\) be the ellipsoid \(x^{2} / 9+y^{2} / 4+z^{2}=1, P\) be the plane \(z=A x+B y,\) and \(C\) be the intersection of \(E\) and \(P\). a. Is \(C\) an ellipse for all values of \(A\) and \(B\) ? Explain. b. Sketch and interpret the situation in which \(A=0\) and \(B \neq 0\). c. Find an equation of the projection of \(C\) on the \(x y\) -plane. d. Assume \(A=\frac{1}{6}\) and \(B=\frac{1}{2} .\) Find a parametric description of \(C\) as a curve in \(\mathbb{R}^{3}\). (Hint: Assume \(C\) is described by \(\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) and find \(a, b, c, d, e, \text { and } f .)\)

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$h(x, y, z)=\sqrt[4]{z^{2}-x z+y z-x y}.$$

A clothing company makes a profit of \(\$ 10\) on its long-sleeved T-shirts and \(\$ 5\) on its short-sleeved T-shirts. Assuming there is a \(\$ 200\) setup cost, the profit on \(\mathrm{T}\) -shirt sales is \(z=10 x+5 y-200,\) where \(x\) is the number of long-sleeved T-shirts sold and \(y\) is the number of short-sleeved T-shirts sold. Assume \(x\) and \(y\) are nonnegative. a. Graph the plane that gives the profit using the window $$ [0,40] \times[0,40] \times[-400,400] $$ b. If \(x=20\) and \(y=10,\) is the profit positive or negative? c. Describe the values of \(x\) and \(y\) for which the company breaks even (for which the profit is zero). Mark this set on your graph.

Use the definition of differentiability to prove that the following functions are differentiable at \((0,0) .\) You must produce functions \(\varepsilon_{1}\) and \(\varepsilon_{2}\) with the required properties. $$f(x, y)=x y$$

In the advanced subject of complex variables, a function typically has the form \(f(x, y)=u(x, y)+i v(x, y),\) where \(u\) and \(v\) are real-valued functions and \(i=\sqrt{-1}\) is the imaginary unit. A function \(f=u+i v\) is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: \(u_{x}=v_{y}\) and \(u_{y}=-v_{x}\) a. Show that \(f(x, y)=\left(x^{2}-y^{2}\right)+i(2 x y)\) is analytic. b. Show that \(f(x, y)=x\left(x^{2}-3 y^{2}\right)+i y\left(3 x^{2}-y^{2}\right)\) is analytic. c. Show that if \(f=u+i v\) is analytic, then \(u_{x x}+u_{y y}=0\) and \(v_{x x}+v_{y y}=0\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Decision 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.