Problem 42 - Calculus an Applied Approach

Question

Identify the quadric surface. $$ \frac{x^{2}}{9}+\frac{y^{2}}{16}+\frac{z^{2}}{16}=1 $$

Step-by-Step Solution

Answer

The given quadric surface is an Ellipsoid.

Step 1: Recognize standard quadric surface equations

The forms of the standard equations for the main types of quadric surfaces are as follows: \n\n Ellipsoid: \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\), \n Hyperboloid of one sheet: \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\), \n Hyperboloid of two sheets: \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\), \n Elliptic paraboloid: \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=z\),\n Hyperbolic paraboloid: \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=z\).

Step 2: Compare the given equation to standard quadric surface equations

On comparing the given equation \( \frac{x^{2}}{9}+\frac{y^{2}}{16}+\frac{z^{2}}{16}=1 \) with the standard equations of quadric surfaces, it is found that it matches the form of the Ellipsoid.

Step 3: Identify the quadric surface

Since the form matches with standard form of an ellipsoid, hence the given quadric surface is an Ellipsoid.

Key Concepts

Ellipsoid

An ellipsoid is a type of quadric surface that resembles a stretched or squished sphere. Imagine a shape that looks like an elongated sphere, similar to a rugby ball or a flattened beach ball. It has three semi-axis lengths, each along the x, y, and z axes, which define its shape. These axes can all be different, leading to stretched forms. An ellipsoid is smooth, closed, and typically symmetric around its center.

For example, the equation\[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1 \]describes an ellipsoid. This equation shows a relationship between the coordinates and the semi-axes lengths (a, b, c). The surface is bounded, meaning it doesn’t extend to infinity, and every point on the surface satisfies this equation.

Ellipsoids have important properties and applications in various fields, like physics and engineering. They help in understanding objects' shapes and can be used to model celestial bodies or even molecules.

Standard Equations

Standard equations are essential in identifying and classifying different quadric surfaces. They provide a formulaic way to recognize geometrical forms by comparing them with known structures.

For the five most common quadric surfaces, the standard equations are:

• Ellipsoid: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1 \)
• Hyperboloid of one sheet: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1 \)
• Hyperboloid of two sheets: \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1 \)
• Elliptic paraboloid: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = z \)
• Hyperbolic paraboloid: \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = z \)

These equations allow for easy recognition and comparison, streamlining the process of identifying three-dimensional shapes. By comparing an equation to these standard forms, one can quickly determine the type of surface described, aiding in visualization and mathematical analysis.

Mathematical Comparison

Mathematical comparison involves lining up a given equation against standard forms to identify the nature of the surface it describes. This is key when working with quadric surfaces and helps establish the characteristics of the surface.

In the provided example, the equation
\[ \frac{x^{2}}{9} + \frac{y^{2}}{16} + \frac{z^{2}}{16} = 1 \]
was compared with standard quadric surfaces. By aligning this equation with the known forms of each type of surface, it's evident that it matches the ellipsoid's standard equation. The coefficients \(a^2, b^2,\) and \(c^2\) are \(9, 16,\) and \(16\) respectively, indicating the radii of the ellipsoid along the x, y, and z axes.

This comparison simplifies the identification process of complex surfaces, turning abstract mathematical expressions into understandable geometric shapes. It's a vital skill in mathematics that aids in effective problem-solving and comprehension.