Problem 57 - Thomas Calculus

Question

Use a CAS to plot the surfaces in Exercises \(53-58 .\) Identify the type of quadric surface from your graph. $$\frac{x^{2}}{9}-1=\frac{y^{2}}{16}+\frac{z^{2}}{2}$$

Step-by-Step Solution

Answer

The surface is a hyperboloid of two sheets.

Step 1: Standard Quadric Surface Form

The given equation is \( \frac{x^{2}}{9}-1=\frac{y^{2}}{16}+\frac{z^{2}}{2} \). We need to rewrite it in a standard quadratic form for easier identification. First, rearrange the equation to get all terms on one side: \( \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2} = 1 \). This is closer to standard form.

Step 2: Identify the Type of Quadric Surface

The standard form \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1 \) represents a hyperboloid of two sheets. In our equation, \( a^2 = 9 \), \( b^2 = 16 \), and \( c^2 = 2 \), which confirms the surface is indeed a hyperboloid of two sheets since there is one positive term (\(x^2\)) and two negative terms (\(y^2\) and \(z^2\)).

Step 3: Using CAS to Plot

Input the rearranged equation \( \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2} = 1 \) into a Computer Algebra System (CAS) to generate the plot. Observe the graph to confirm the shape corresponds to a hyperboloid of two sheets. The plot should show two separate surfaces centered around the x-axis.

Key Concepts

Hyperboloid of Two Sheets

A hyperboloid of two sheets is a fascinating three-dimensional surface that can be visualized as two separate pieces or 'sheets'. To recognize this type of quadric surface, note that it involves a quadratic equation with one positive term and two negative terms. This is different from a hyperboloid of one sheet, where two terms are positive and one is negative. The equation given in the problem, \( \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2} = 1 \), is a perfect example of a hyperboloid of two sheets. Here:

• The positive term \( \frac{x^{2}}{9} \) shapes the axis and orientation.
• The negative terms \( \frac{y^{2}}{16} \) and \( \frac{z^{2}}{2} \) indicate the sheets spread away from the x-axis.

Hyperboloids of two sheets are not physically connected, and they appear in certain architectural structures and theoretical physics. If you're plotting this, expect to see two symmetric, bowl-like shapes facing away from each other along the x-axis.

Computer Algebra System

A Computer Algebra System (CAS) is a powerful tool for solving complex mathematical problems, including plotting intricate graphs like quadric surfaces. In this context, a CAS helps students and professionals visualize and analyze multivariable equations, which are otherwise hard to plot manually.
A CAS can:

• Quickly rearrange equations into standard forms for better understanding.
• Efficiently graph three-dimensional surfaces with accuracy.
• Allow manipulation and exploration of mathematical models with interactive tools.

For example, when you input the equation \( \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2} = 1 \) into a CAS, it will generate a 3D plot. This helps in identifying the shape, orientation, and other features of the hyperboloid of two sheets. Such technology makes learning and problem-solving more engaging and less tedious.

Standard Form of Quadrics

Understanding the standard form of quadrics is essential for identifying and analyzing various surfaces. Quadrics are any surfaces described by a polynomial equation of degree two in three variables. They include ellipsoids, paraboloids, and hyperboloids among others.
For a hyperboloid of two sheets, the standard form is:\[ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1 \]This form makes it easier to analyze the surface properties like:

• Orientation: Which axis the surface aligns with.
• Shape: Two disconnected surfaces when \( \frac{x^{2}}{a^{2}} \) is positive and others are negative.
• Dimensions: Defined by \( a^2, b^2, \text{ and } c^2 \).

Using the standard form, students can quickly identify surface types and their properties. It helps in comparing different quadric surfaces and understanding their real-world applications. Recognizing these equations in standard form simplifies the learning process and aids in better comprehension during problem-solving and graph visualization.