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Magazine Chapter 12: Problem 43
Sketch the surfaces in Exercises \(13-44\) . ASSORTED $$ 4 y^{2}+z^{2}-4 x^{2}=4 $$
Short Answer
Expert verified
The surface is a hyperboloid of one sheet opening along the x-axis.
Step by step solution
01
Identify the Type of Surface
The given equation is \( 4y^2 + z^2 - 4x^2 = 4 \). Comparing this with the standard form of a hyperboloid \( abla_1x^2 + abla_2y^2 + abla_3z^2 = C \), we see that it matches the standard form of a hyperboloid of one sheet: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \).
02
Simplify the Equation
Divide every term in the equation \( 4y^2 + z^2 - 4x^2 = 4 \) by 4 to simplify it: \( y^2 + \frac{z^2}{4} - x^2 = 1 \).
03
Identify Axes and Transverse Axis
From \( y^2 + \frac{z^2}{4} - x^2 = 1 \), recognize that the minus sign in front of \( x^2 \) indicates the hyperboloid opens along the x-axis, making the x-axis the transverse axis. The plus terms \( y^2 \) and \( \frac{z^2}{4} \) indicate ellipses in the yz-plane.
04
Determine Cross-Sections
For a fixed value of x, the equation \( y^2 + \frac{z^2}{4} = 1 + x^2 \) is an ellipse in the yz-plane. The size of the ellipse increases as \( x^2 \) increases because \( 1 + x^2 \) becomes larger.
05
Sketch the Surface
Begin sketching by plotting the central hyperbola by setting \( x=0 \), resulting in the ellipse \( y^2 + \frac{z^2}{4} = 1 \). As you move along the x-axis, the ellipses increase in size due to the addition of \( x^2 \). Draw symmetric ellipses along the x-axis on both positive and negative sides, indicating a central opening at the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ellipses
Ellipses are elongated circles formed by stretching a circle along one or more axes. In the context of the given equation, ellipses appear in the yz-plane when we hold the value of x constant.
The equation you observe here, with terms like \( y^2 + \frac{z^2}{4} = 1 + x^2 \), illustrates how these ellipses change as x varies.
The equation you observe here, with terms like \( y^2 + \frac{z^2}{4} = 1 + x^2 \), illustrates how these ellipses change as x varies.
- The term \( y^2 \) represents the vertical axis of the ellipse.
- The term \( \frac{z^2}{4} \) represents the horizontal axis.
Transverse Axis
The transverse axis is pivotal in defining the direction a hyperboloid opens. In our context, it identifies the primary axis along which the hyperboloid expands.
For our example, the original equation \( 4y^2 + z^2 - 4x^2 = 4 \) converted to \( y^2 + \frac{z^2}{4} - x^2 = 1 \), reveals the transverse axis.
For our example, the original equation \( 4y^2 + z^2 - 4x^2 = 4 \) converted to \( y^2 + \frac{z^2}{4} - x^2 = 1 \), reveals the transverse axis.
- The negative sign before \( x^2 \) indicates the hyperboloid opens along the x-axis.
- This axis essentially dictates the orientation and growth direction of the hyperboloid, signifying which axis the structure extends around.
Cross-Sections
Cross-sections of a shape are essentially slices or intersections that provide insight into its three-dimensional structure.
In the equation \( y^2 + \frac{z^2}{4} - x^2 = 1 \), each specific value of x gives us a different cross-section.
In the equation \( y^2 + \frac{z^2}{4} - x^2 = 1 \), each specific value of x gives us a different cross-section.
- Each of these is represented by the equation \( y^2 + \frac{z^2}{4} = 1 + x^2 \).
- These cross-sections are ellipses, and their size grows along the transverse axis as \( x^2 \) increases.
Standard Form of Hyperboloid
The standard form of a hyperboloid defines its structural and mathematical essence, crucial for analyzing and sketching its shape.
The equation given \( 4y^2 + z^2 - 4x^2 = 4 \), upon simplification, is transformed into \( y^2 + \frac{z^2}{4} - x^2 = 1 \).
The equation given \( 4y^2 + z^2 - 4x^2 = 4 \), upon simplification, is transformed into \( y^2 + \frac{z^2}{4} - x^2 = 1 \).
- This simplified form corresponds to a hyperboloid of one sheet with the template \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \).
- The presence of different terms for \( y^2 \), \( z^2 \), and \( x^2 \) coefficients indicates which axis is elongated and how the surface stretches along axes.
