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Chapter 14: Problem 4

Sketch the graph of the cylinder in an \(x y z-\) coordinate system. $$x^{2}+5 z^{2}=25$$

Short Answer

Expert verified
It's an infinite cylinder along the y-axis with elliptical cross-sections in the xz-plane.

Step by step solution

01

Identify the Shape

The equation given is \[x^2 + 5z^2 = 25.\]Recognize this as a standard form of an equation of a cylinder involving two variables, in this case, \(x\) and \(z\). Since \(y\) is not present, this implies that the cylinder extends infinitely in the \(y\)-direction.
02

Rewrite in Standard Form

We want to express the equation in a form where it's easier to identify the type of conic section. The equation in standard form is \[\frac{x^2}{25} + \frac{z^2}{5} = 1.\] This is similar to an ellipse equation in the \(xz\)-plane.
03

Interpret the Cross Section

Consider the cross-section of the cylinder. By setting \(y = 0\), we analyze the \(xz\)-plane: \[x^2 + 5z^2 = 25.\] This is an ellipse with the semi-major axis along the \(x\)-axis (radius 5) and semi-minor axis along the \(z\)-axis (radius \(\sqrt{5}\)).
04

Sketch the Ellipse

Draw the ellipse in the \(xz\)-plane based on the axes sizes identified: horizontal radius \(5\) along the \(x\)-axis, and a vertical radius of \(\sqrt{5}\) along the \(z\)-axis.
05

Extend along the Y-axis

Because \(y\) is not present in the equation, the complete graph of this equation in 3D is obtained by extending the ellipse in the \(xz\)-plane infinitely along the \(y\)-axis, resulting in an infinite cylinder.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3D graphing
Three-dimensional graphing allows us to visually represent mathematical equations or objects in a coordinate system that includes three axes: the x-axis, y-axis, and z-axis. In the given problem, we're dealing with a three-dimensional object, specifically a cylinder. The equation \( x^2 + 5z^2 = 25 \) represents a standard quadratic form. However, understanding it in a 3D space requires identifying how it behaves relative to all three axes.
  • The equation uses two variables, \(x\) and \(z\), without \(y\), meaning the object repeats or extends along the \(y\)-axis.
  • This repetition characterizes a cylinder, which extends infinitely in one of the 3D dimensions.
By placing and drawing this object in a 3D system, we observe how it behaves parallel and perpendicular to the axes, offering a richer visualization of mathematical properties in space.
ellipse cross-section
A cross-section refers to the intersection of a three-dimensional figure with a plane. For this cylinder, the cross-section can be analyzed by fixing one dimension, such as \(y = 0\). This allows us to examine the shape in the \(xz\)-plane, which resembles an ellipse.
  • In the equation \( x^2 + 5z^2 = 25 \), setting \( y = 0 \) highlights the elliptical nature along \(x\) and \(z\).
  • It results in an ellipse with semi-major axis (5 units) along the \(x\)-axis and semi-minor axis (\(\sqrt{5}\) units) along the \(z\)-axis.
Understanding this helps in visualizing the shape of the cylinder's cross-section and the dimensions of its components.
infinite cylinder
An infinite cylinder in 3D refers to a surface extending endlessly in one direction. In this case, since \(y\) is absent from \( x^2 + 5z^2 = 25 \), the cylinder is longitudinally infinite in the \(y\)-axis direction.
  • Without a boundary in the y-direction, the cylinder does not have a finite length but stretches infinitely.
  • The structure is based on the repeating elliptical cross-sections perpendicular to the \(y\)-axis.
This concept of infinities in certain dimensions is key in mathematics, indicating behaviors and patterns that continue without limit, aiding in abstract analysis and real-world applications like pipe flow and structural engineering.
conic sections
Conic sections are curves obtained by slicing a double cone with a plane. Common examples include ellipses, parabolas, and hyperbolas. The equation \(x^2 + 5z^2 = 25\) depicts an elliptical section.
  • Here, slices parallel to the base (in the \(xz\)-plane) are elliptical, as evidenced by the transformation into the form \(\frac{x^2}{25} + \frac{z^2}{5} = 1\).
  • These elliptical sections are characteristic of the cylinder's mathematical structure.
By studying the conic sections involved, one can abstract the geometrical features of complex three-dimensional shapes and understand the foundational aspects of curves that define them.

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Most popular questions from this chapter

Graph \(f\) and \(g\) on the same coordinate plane for \(-2 \leq x \leq 2\). (a) Estimate the coordinates of their point \(P\) of intersection. (b) Approximate the angles between the tangent lines to the graphs at \(P\). $$ f(x)=1-3 x+x^{3}, \quad g(x)=x^{5}+\frac{1}{2} $$

Sketch the graph of the quadric surface. Paraboloids (a) \(z=x^{2}+\frac{y^{2}}{9}\) (b) \(\frac{z^{2}}{25}+\frac{y^{2}}{9}-x=0\)

Sketch the graph of the equation in an \(x y z-\) coordinate system, and identify the surface. $$y^{2}-9 x^{2}-z^{2}-9=0$$

Sketch the graph of the equation in an \(x y z-\) coordinate system, and identify the surface. $$2 x+4 y+3 z=12$$

Solve without using components for the vectors. In the Cauchy-Schwarz inequality, under what conditions is \(|\mathbf{a} \cdot \mathbf{b}|=\|\mathbf{a}\|\|\mathbf{b}\| ?\)
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