/*! 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 30 Find the equation of the surface... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 30

Find the equation of the surface that results when the curve \(4 x^{2}-3 y^{2}=12\) in the \(x y\)-plane is revolved about the \(x\)-axis.

Short Answer

Expert verified
The surface equation is \( 4x^2 - 3(y^2 + z^2) = 12 \).

Step by step solution

01

Identifying the Original Curve

The given curve is an equation of a hyperbola: \(4x^2 - 3y^2 = 12\). This is a hyperbola with its transverse axis along the \(x\)-axis in the \(xy\)-plane.
02

Re-writing the Equation for Simplification

Rearrange the original hyperbola equation to standard form by dividing everything by 12:\[ \frac{x^2}{3} - \frac{y^2}{4} = 1. \]
03

Understanding the Revolution

When the curve is revolved around the \(x\)-axis, each point on the curve traces out a circle in space with a radius equal to the \(y\)-coordinate of the original point.
04

Forming the Surface Equation Upon Revolution

By revolving about the \(x\)-axis, replace \(y\) with \( \sqrt{y^2 + z^2} \) in the equation to account for the circular cross-section:\[ \frac{x^2}{3} - \frac{y^2 + z^2}{4} = 1. \]
05

Final Equation of the Surface

The surface formed is an equation: \[ 4x^2 - 3(y^2 + z^2) = 12. \] This is the required equation representing the surface formed when the hyperbola is revolved around the \(x\)-axis.

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.

Equation of a surface
In calculus, the equation of a surface is a mathematical expression that describes all the points lying on a geometric surface in three-dimensional space. A surface can be seen as an extension of two-dimensional curves into an extra dimension.
For this exercise, we start with a hyperbola, which is a type of conic section represented by the equation: \(4x^2 - 3y^2 = 12\).
We are tasked with finding the three-dimensional surface resulting from revolving this hyperbola around the \(x\)-axis.
To accomplish this, you convert the two-dimensional equation of the curve into a three-dimensional surface equation.
  • First, it is useful to restate the hyperbola in its standard form to make subsequent steps more straightforward: \(\frac{x^2}{3} - \frac{y^2}{4} = 1\).
  • Next, since we are revolving the curve about the \(x\)-axis, we incorporate an additional variable, \(z\), to transform the original curve into a surface.
Hyperbola
A hyperbola is a type of conic section formed by the intersection of a plane with both nappes of a double cone. It appears as two separate, symmetrical curves.
Every hyperbola has certain characteristics:
  • Two branches that open either vertically or horizontally depending on its orientation.
  • A center, which is the point equidistant from foci of the hyperbola.
  • Asymptotes, which are straight lines that the branches approach but never touch.
  • A transverse axis, the line segment joining the vertices.
In our exercise, the original equation \(4x^2 - 3y^2 = 12\) represents a hyperbola in the \(xy\)-plane with its transverse axis along the \(x\)-axis.
Rearranging it to the standard form: \(\frac{x^2}{3} - \frac{y^2}{4} = 1\), we can more clearly identify its properties. This hyperbola is centered at the origin and its transverse axis is parallel to the \(x\)-axis.
Revolution around an axis
When a curve is revolved around an axis, it creates a surface in three-dimensional space. This process involves rotating every point on the curve by 360 degrees around the axis.
The original problem requires us to revolve a hyperbola about the \(x\)-axis.
  • The revolution converts each point \((x, y)\) on the hyperbola into a circle of radius equal to \(y\), lying in a plane perpendicular to the axis of revolution.
  • In mathematical terms, this is achieved by replacing the \(y\) coordinate in the hyperbola equation with \(\sqrt{y^2 + z^2}\), which accounts for all the points in the circular cross-section.
Carrying out this transformation gives us the surface equation: \(4x^2 - 3(y^2 + z^2) = 12\).
This equation describes a hyperboloid of one sheet, which is the three-dimensional surface obtained after revolving the hyperbola around the \(x\)-axis.
This elegant surface depicts a symmetric shape extending in three dimensions, perfectly illustrating how calculus can transform simple curves into complex geometrical structures.

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 \(\mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle\) and \(\mathbf{b}=\left\langle b_{1}, b_{2}, b_{3}\right\rangle\) be fixed vectors. Show that \((\mathbf{x}-\mathbf{a}) \cdot(\mathbf{x}-\mathbf{b})=0\) is the equation of a sphere, and find its center and radius.

A wind with velocity 45 miles per hour is blowing in the direction \(\mathrm{N} 20^{\circ} \mathrm{W}\). An airplane that flies at 425 miles per hour in still air is supposed to fly straight north. How should the airplane be headed and how fast will it then be flying with respect to the ground?

Show that the work done by a constant force \(\mathbf{F}\) on an object that moves completely around a closed polygonal path is 0 .

Suppose that the three coordinate planes bounding the first octant are mirrors. A light ray with direction \(a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) is reflected successively from the \(x y\)-plane, the \(x z\)-plane, and the \(y z\)-plane. Determine the direction of the ray after each reflection, and state a nice conclusion concerning the final reflected ray.

Find the equation of the plane through \((-1,-2,3)\) and perpendicular to both the planes \(x-3 y+2 z=7\) and \(2 x-2 y-z=-3\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.