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Chapter 13: Problem 51

(a) Graph the curve with parametric equations $$ \begin{array}{l}{x=\frac{27}{26} \sin 8 t-\frac{8}{39} \sin 18 t} \\\ {y=-\frac{27}{26} \cos 8 t+\frac{8}{39} \cos 18 t} \\ {z=\frac{144}{65} \sin 5 t}\end{array} $$ $$ \begin{array}{l}{\text { (b) Show that the curve lies on the hyperboloid of one sheet }} \\ {144 x^{2}+144 y^{2}-25 z^{2}=100 \text { . }}\end{array} $$

Short Answer

Expert verified
The curve lies on the hyperboloid after substituting into the equation and simplifying.

Step by step solution

01

Analyze the Parametric Equations

The given parametric equations are for coordinates \( x \), \( y \), and \( z \) in terms of the parameter \( t \). Each equation is a combination of sine or cosine functions with different frequencies. Notice that the \( x \) and \( y \) coordinates are expressed with 8 and 18 times the parameter \( t \), and \( z \) is expressed with 5 times the parameter \( t \).
02

Graph the Parametric Curve

Using a software or graphing tool, plot the parametric equations: \( x(t) = \frac{27}{26} \sin(8t) - \frac{8}{39} \sin(18t) \), \( y(t) = -\frac{27}{26} \cos(8t) + \frac{8}{39} \cos(18t) \), \( z(t) = \frac{144}{65} \sin(5t) \). Set \( t \) over a reasonable interval, such as \( 0 \) to \( 2\pi \), to view the full shape of the curve.
03

Compute \( x^2 + y^2 \)

To show that the curve lies on a hyperboloid, we will express \( x^2 + y^2 \). Simplify \( x^2 \) and \( y^2 \):\[ x^2 = \left(\frac{27}{26} \sin 8t - \frac{8}{39} \sin 18t \right)^2 \] and \[ y^2 = \left(-\frac{27}{26} \cos 8t + \frac{8}{39} \cos 18t \right)^2 \]. Then calculate \( x^2 + y^2 \).
04

Simplify \( x^2 + y^2 \) and Compare to the Hyperboloid Equation

After expanding and simplifying \( x^2 + y^2 \), insert this into the hyperboloid equation \( 144 x^2 + 144 y^2 - 25 z^2 = 100 \). Substitute \( z = \frac{144}{65} \sin 5t \) and simplify to demonstrate that this equation holds true for the curve. This confirms that the curve lies on the hyperboloid.

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Key Concepts

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

Parametric Curves
Parametric curves are an exciting part of mathematics that describe curves by expressing their coordinates as functions of a parameter, often denoted as \( t \). This approach allows for a more flexible description of curves, making it possible to explore more complex shapes that aren't easily captured by standard Cartesian equations.
  • In our exercise, the parametric equations define the coordinates \( x \), \( y \), and \( z \) in terms of \( t \).
  • The functions \( x(t) = \frac{27}{26} \sin(8t) - \frac{8}{39} \sin(18t) \) and \( y(t) = -\frac{27}{26} \cos(8t) + \frac{8}{39} \cos(18t) \) describe the curve's movement across the plane as \( t \) advances.
  • The function \( z(t) = \frac{144}{65} \sin(5t) \) adds a vertical component, showcasing the curve in three-dimensional space.
Each equation uses trigonometric functions of different frequencies, allowing for the creation of a complex and intertwined path. As \( t \) sweeps through its range, these curves unfold their shapes, illustrating their beauty. By studying these equations, we can capture the essence of intricate three-dimensional paths.
Hyperboloid
A hyperboloid is a type of quadric surface that can be visualized as a surface generated by rotating a hyperbola around one of its principal axes. It comes in two types: one-sheet and two-sheet hyperboloids. Our focus is the hyperboloid of one sheet.
  • This surface is characterized by the equation \( 144x^2 + 144y^2 - 25z^2 = 100 \).
  • The curve described by our parametric equations lies on this surface, meaning every point on the curve satisfies the hyperboloid equation.
To understand why this holds, note the interaction between \( x^2, y^2, \) and \( z^2 \) when plugged into the hyperboloid equation. Parametric equations naturally frame together the defined curve within the larger structure of this surface, illustrating a profound interaction between algebraic surfaces and curves. When modeling, the hyperboloid's structure offers intuitive pathways to explore its intriguing geometry.
Trigonometric Functions
Trigonometric functions, like sine and cosine, play a key role in parametric equations, often controlling the periodic nature of these curves. These functions repeat their values in regular intervals, providing uniformity and structure across cyclic behaviors.
  • The sine function, as seen in \( x(t) \) and \( z(t) \), contributes to oscillating movements along the \( x \) and \( z \)-axes.
  • Cosine functions in \( y(t) \) mirror this periodic behavior but oriented orthogonally to sine's path, hence adding depth with cosine movement.
  • The parametric equations mix frequencies of \( 8 \), \( 18 \), and \( 5 \) for \( x, y, \) and \( z \) respectively, utilizing frequency changes to create complex patterns.
These functions are powerful tools in mathematical modeling, especially when defining cyclic or wave-like structures. The angles, defined by parameter \( t \), wield trigonometric functions to sculpt the path of the curve beautifully. The harmony of sine and cosine helps forge an aesthetic and mathematically robust representation of rotational and oscillatory paths.
Three-Dimensional Graphing
Three-dimensional graphing involves plotting points in a 3D space to visualize geometric properties and interactions of curves and surfaces. Understanding this method is crucial for interpreting complex equations.
  • The parametric curve from our exercise is plotted in 3D, requiring a graphing tool that can handle complex trigonometric expressions.
  • By setting \( t \) across a suitable range, such as \( 0 \) to \( 2\pi \), you can observe the entire curve as it unfolds over this interval.
  • This visualization aids in comprehending spatial relations and the nature of the curve's path as it constrains itself onto the hyperboloid.
Graphing software translates abstract parametric equations into tangible visuals. Through 3D graphing, students can gain insight and appreciation for the spatial dynamics at play, providing a deeper understanding of three-dimensional geometry and its capabilities in visualization.

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Most popular questions from this chapter

A medieval city has the shape of a square and is protected by walls with length \(500 \mathrm{m}\) and height \(15 \mathrm{m}\). You are the commander of an attacking army and the closest you can get to the wall is \(100 \mathrm{m}\). Your plan is to set fire to the city by catapulting heated rocks over the wall (with an initial speed of \(80 \mathrm{m} / \mathrm{s}\) ). At what range of angles should you tell your men to set the catapult? (Assume the path of the rocks is perpendicular to the wall.)

Find the unit tangent vector \(\mathbf{T}(t)\) at the point with the given value of the parameter \(t\) $$ \mathbf{r}(t)=\left\langle t^{2}-2 t, 1+3 t, \frac{1}{3} t^{3}+\frac{1}{2} t^{2}\right\rangle, \quad t=2 $$

Find the derivative of the vector function. $$ \mathbf{r}(t)=t \mathbf{a} \times(\mathbf{b}+t \mathbf{c}) $$

The following formulas, called the Frenet-Serret formulas, are of fundamental importance in differential geometry: $$ \begin{array}{l}{\text { 1. } d \mathbf{T} / d s=\kappa \mathbf{N}} \\ {\text { 2. } d \mathbf{N} / d s=-\kappa \mathbf{T}+\tau \mathbf{B}} \\ {\text { 3. } d \mathbf{B} / d s=-\tau \mathbf{N}}\end{array} $$ $$ \begin{array}{l}{\text { (Formula I comes from Exercise } 59 \text { and Formula } 3 \text { comes }} \\ {\text { from Exercise } 61 . \text { ) Use the fact that } \mathbf{N}=\mathbf{B} \times \mathbf{T} \text { to }} \\\ {\text { deduce Formula } 2 \text { from Formulas } 1 \text { and } 3 .}\end{array} $$

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$ x=e^{-t} \cos t, \quad y=e^{-t} \sin t, \quad z=e^{-t} ; \quad(1,0,1) $$
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