/*! 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 41 Sketch the quadric surface. \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 41

Sketch the quadric surface. \(y^{2}+z^{2}=9\)

Short Answer

Expert verified
It's a circular cylinder along the x-axis with radius 3 in the yz-plane.

Step by step solution

01

Recognize the Equation Type

The equation given is in the form of \[ y^2 + z^2 = 9 \]This is the standard form of a cylinder, specifically a circular cylinder, because it only involves the variables \(y\) and \(z\). The term missing is the \(x\) variable, implying the circular cross-section's invariance along the \(x\)-axis.
02

Identify the Shape of Cross-sections

Examine the equation \[ y^2 + z^2 = 9 \]This form represents a circle with radius 3 in the \(yz\)-plane (axes \(y\) and \(z\)). Every cross-section parallel to the \(yz\)-plane is a circle of radius 3.
03

Determine the Direction of the Cylinder

Since the equation does not involve the variable \(x\), the surface described is a cylinder that extends infinitely along the \(x\)-axis. Therefore, for every value of \(x\), the cross-section is the same circle.
04

Sketch the Surface

In the sketch, draw the cross-section circle with radius 3 in the \(yz\)-plane centered at the origin \((y, z) = (0, 0)\). Then replicate these circles along the \(x\)-axis in both directions, showing that the cylinder is extended along the \(x\)-axis. The result is an infinitely long cylinder with circular cross-sections.

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.

Circular Cylinder
A circular cylinder is a three-dimensional surface formed by moving a circle in a straight line through space. In the context of the equation \( y^2 + z^2 = 9 \), this movement occurs along the \(x\)-axis, since the equation lacks an \(x\) term. The essence of a circular cylinder can be described as:
  • The circle remains constant in shape and size; this one has a radius of 3.
  • The circles stack infinitely along the given direction (here, the \(x\)-axis).
This concept is useful because it lets us visualize complex 3D objects through a simple repeated shape. Unlike other surfaces that might twist or curve more dramatically, circular cylinders maintain the same cross-section all the way through.
Cross-sections
Cross-sections are slices or views of a three-dimensional object taken parallel to a plane. For the equation \( y^2 + z^2 = 9 \), each cross-section in the \(yz\)-plane is crucial to understanding the surface. What you'll see is a circle:
  • Each circle has a radius of 3.
  • The center of these circles is at the origin, \((0,0)\), on the \(yz\)-plane.
This means all visual slices of this cylinder, taken parallel to the \(yz\)-plane, appear identical. Each circle implies a perfectly round cut through the cylinder, aiding in picturing how the cylinder extends through space along the \(x\)-axis. Recognizing these cross-sections allows us to grasp the full 3D shape by understanding just one consistent two-dimensional form.
Equation Type
The equation \( y^2 + z^2 = 9 \) belongs to a specific group of equations that describe quadric surfaces. Quadric surfaces are second-degree equations in three variables (\(x\), \(y\), and \(z\)). Despite their variety, the absence of one variable simplifies interpretation:
  • The absence of the \(x\) term indicates that the surface extends along the \(x\)-axis.
  • The remaining equation, \( y^2 + z^2 = 9 \), depicts a set of circles.
This particular equation is known as a circular cylinder. The consistent structure of the equation, involving only the squares of two variables, reveals two key aspects: the shape of cross-sections (circles) and the direction of extension. Recognizing the equation type simplifies the process of sketching and understanding its geometry.

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 \(f(x, y)=A x^{2}+2 B x y+C y^{2}\), as in (1). Assume that \(A \neq 0\) and \(A C-B^{2}<0\). Verify that there are points \((x, y)\) as close to \((0,0)\) as one wishes such that \(f(x, y)>0\), and other points \((x, y)\) as close to \((0,0)\) as one wishes such that \(f(x, y)<0 .\) Conclude that \(f\) has a saddle point at \((0,0) .\) (Hint: Consider points of the form \((x, 0)\) and \((-B y / A, y) .)\)

Use implicit differentiation to find \(\partial z / \partial x\) and \(\partial z / \partial y\) at the given point. Then find an equation of the plane tangent to the level surface at that point. $$ x^{2}-y^{2}-z^{2}=1 ;(\sqrt{2}, 0,1) $$

Find the minimum volume of a tetrahedron in the first octant bounded by the planes \(x=0, y=0, z=0\), and \(a\) plane tangent to the sphere \(x^{2}+y^{2}+z^{2}=1\). (Hint: If the plane is tangent to the sphere at the point \(\left(x_{0}, y_{0}, z_{0}\right)\), then the volume of the tetrahedron is \(1 /\left(6 x_{0} y_{0} z_{0}\right)\).)

Suppose that on your vacation you plan to spend \(x\) days in San Francisco, \(y\) days in your home town, and \(z\) days in New York. You calculate that your total enjoyment \(f(x, y, z)\) will be given by $$ f(x, y, z)=2 x+y+2 z $$ If plans and financial limitations dictate that $$ x^{2}+y^{2}+z^{2}=225 $$ how long should each stay be to maximize your enjoyment?

A metal plate with vertices \((1,1),(5,1),(1,3)\), and \((5,3)\) is heated by a flame at the origin, and the temperature at a point on the plate is inversely proportional to the distance from the origin. If an ant is located at the point \((3,2)\), in what direction should the ant crawl to cool the fastest?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.