Problem 57 - University Calculus: Early Transcendentals

Question

Use a CAS to plot the surfaces in Exercises. 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 graph is a hyperboloid of one sheet.

Step 1: Identify the Standard Form of the Quadric Surface

The given equation \( \frac{x^{2}}{9} - 1 = \frac{y^{2}}{16} + \frac{z^{2}}{2} \) can be rewritten by adding 1 to all terms on the right side: \( \frac{x^{2}}{9} = \frac{y^{2}}{16} + \frac{z^{2}}{2} + 1 \). By adding 1 to the left side as well, the equation becomes \( \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2} = 1 \), fitting the standard form of a hyperboloid of one sheet \( \frac{x^{2}}{a^2} - \frac{y^{2}}{b^2} - \frac{z^{2}}{c^2} = 1 \).

Step 2: Set Up Computational Algebra System (CAS)

Input this final equation \( f(x, y, z) = \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2} \) into a CAS such as GeoGebra, Mathematica, or Desmos to graphically represent the surface.

Step 3: Plot the Graph Using CAS

Use the CAS to render the 3D plot of the surface. Ensure the correct range is set for \(x\), \(y\), and \(z\) to capture the shape adequately. Adjust the viewing angles to better visualize the characteristics of the surface.

Step 4: Analyze the Graph and Identify the Surface

Examine the plotted surface. A hyperboloid of one sheet will typically appear as a saddle-shaped surface extending along the axis of the lone positive term, in this case, the \(x\)-axis. Confirm this shape is present by observing the graph.

Key Concepts

Hyperboloid of One Sheet

A hyperboloid of one sheet is a type of quadric surface. These surfaces are characterized by their unique geometric shape, which looks somewhat like a distorted cylinder or an hourglass. To identify this surface, we look for the standard form equation:\[ \frac{x^{2}}{a^2} - \frac{y^{2}}{b^2} - \frac{z^{2}}{c^2} = 1 \]In this form, there is one positive term and two negative terms on the left side of the equation. It is important to notice the position of the positive term as it indicates the direction along which the surface will open, stretching out like an elongated saddle or horn. This unique structure allows hyperboloids of one sheet to extend infinitely along their main axis. In our exercise, this axis corresponds to the \(x\)-axis, where the \(x^2\) term is positive. Recognizing this pattern can help you quickly identify a hyperboloid of one sheet when analyzing new equations.

Computational Algebra Systems

Computational Algebra Systems (CAS) are powerful tools that help in visualizing complex mathematical concepts, such as quadric surfaces. CAS like GeoGebra, Mathematica, and Desmos allow users to input equations and render them as 3D plots, providing a clear visual representation. Using CAS makes it easier to understand the real-world geometry of mathematical objects. They calculate positions, intersections, and shapes at a high level of precision, which can be challenging by hand. A typical process involves:

• Inputting the standard form equation into the CAS: something like \(f(x, y, z) = \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2}\).
• Allowing the system to compute the 3D plot, highlighting the unique structure of the surface.
• Adjusting scale and view settings to emphasize crucial aspects of the graph.

These systems can significantly enhance comprehension, especially for visual learners, by providing a direct link between theoretical equations and their geometric representations.

3D Plotting

3D plotting is an essential technique in gaining insights into how mathematical surfaces behave in space. When you plot in three dimensions, you provide a visual image of equations that may otherwise be tough to imagine. In 3D plotting, it is crucial to consider:

• The correct range for all variables. This ensures you capture the entire surface without any clipping or stretching.
• Adjusting the viewing angle to pick up on subtle surface details like curves, twists, and intersections.
• Using labels and annotations to mark significant features or axes.

The value of 3D plots shines when dealing with complex surfaces like hyperboloids of one sheet. These plots reveal the surface's saddle shape and help you assess its dimensions and angles better than flat, 2D representations.

Equation Standard Forms

Understanding standard forms of equations is a staple in identifying and working with quadric surfaces. The standard form categorizes surfaces, assists in recognizing their geometric properties, and guides how we interpret their behavior.For hyperboloid equations, the standard form is:\[ \frac{x^{2}}{a^2} - \frac{y^{2}}{b^2} - \frac{z^{2}}{c^2} = 1 \]In this expression:

• The positive term dictates the direction the surface extends towards.
• The negative terms indicate compressions that give the surface its saddle-like shape.

By transforming an equation into its standard form, you not only simplify its expression but also unlock the ability to interpret its geometrical and graphical properties more effectively. Recognizing a hyperboloid from its equation becomes second nature when you are familiar with these standard form signals.