Problem 93 - Thomas Calculus

Question

Use a CAS to plot the surfaces in Exercises \(89-94\) . 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 one sheet.

Step 1: Rearrange the Equation

Rearrange the given equation to resemble a standard form for a quadric surface. The given equation is: \( \frac{x^{2}}{9}-1=\frac{y^{2}}{16}+\frac{z^{2}}{2} \). By moving terms to one side, it becomes \( \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2} = 1 \).

Step 2: Recognize the Standard Form

The rearranged equation \( \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2} = 1 \) can be compared with the standard form of a hyperboloid of one sheet: \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1 \). This shows that the equation represents a hyperboloid of one sheet.

Step 3: Use a CAS Tool

Input the equation \( \frac{x^{2}}{9} - \frac{y^{2}}{16} - \frac{z^{2}}{2} = 1 \) into a Computer Algebra System (CAS) to generate a 3D plot. Ensure that the plot covers a range of values for \(x, y,\) and \(z\) to see the complete shape of the surface.

Step 4: Analyze the Plot

Examine the 3D plot generated by the CAS to confirm the characteristics of a hyperboloid of one sheet. Look for the defining features such as an hourglass shape or a continuous surface that does not have breaks or separate components.

Key Concepts

Hyperboloid of One Sheet

A hyperboloid of one sheet is a fascinating quadric surface that resembles an hourglass shape. It is defined by an equation of the form: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \]. The surface is composed of one continuous, unbroken sheet. This means there are no disconnected parts, unlike some other quadric surfaces.

Key characteristics of a hyperboloid of one sheet include:

• An hourglass or saddle shape.
• Symmetry around the central axis – which in this case is the x-axis.
• Curves in distinct directions, which makes it both complex and beautiful.

As we have seen in the rearranged equation \( \frac{x^2}{9} - \frac{y^2}{16} - \frac{z^2}{2} = 1 \), the coefficients indicate different stretches in x, y, and z directions.

Computer Algebra System

A Computer Algebra System (CAS) is a powerful tool used to manipulate mathematical equations and visualize complex surfaces like our hyperboloid of one sheet.

Key features of a CAS include:

• Ability to simplify, rearrange, and solve complex mathematical expressions.
• Generate accurate graphical representations for better understanding.
• Offer insights into the behavior of mathematical surfaces in three-dimensional space.

In our exercise, using a CAS to input the equation \( \frac{x^2}{9} - \frac{y^2}{16} - \frac{z^2}{2} = 1 \) delivers a 3D plot. This invaluable visual aid helps identify characteristics unique to hyperboloid of one sheet.

3D Plotting

3D plotting is an excellent method to visualize complex surfaces. It helps in understanding and analyzing the shapes and properties of mathematical equations visually.

When plotting the equation \( \frac{x^2}{9} - \frac{y^2}{16} - \frac{z^2}{2} = 1 \) using a CAS:

• You provide ranges for x, y, and z to see the full form of the surface.
• The plot will show the hyperboloid's distinct hourglass shape.
• Rotating and zooming allows detailed inspection of symmetry and geometry.

3D plotting aids in recognizing not just hyperboloids, but various quadric surfaces, making complex mathematical concepts much more approachable.

Standard Form of Quadric Surfaces

The standard form of quadric surfaces simplifies the identification and classification of surfaces in three-dimensional space. Each type of quadric has a unique form.

Key types based on their standard forms include:

• Ellipsoids (\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]).
• Hyperboloids (One Sheet & Two Sheets, as with:\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \] for one sheet).
• Paraboloids (Elliptic and Hyperbolic).

By rearranging the given equation into the standard form, as shown in the solution, we were able to efficiently identify the quadric surface as a hyperboloid of one sheet.