Problem 53 - University Calculus: Early Transcendentals

Question

Use a CAS to plot the surfaces in Exercises. Identify the type of quadric surface from your graph. $$\frac{x^{2}}{9}+\frac{y^{2}}{36}=1-\frac{z^{2}}{25}$$

Step-by-Step Solution

Answer

The quadric surface is an ellipsoid.

Step 1: Rewrite the Equation

First, we rewrite the given equation: \( \frac{x^{2}}{9} + \frac{y^{2}}{36} = 1 - \frac{z^{2}}{25} \). To identify the surface, we can rearrange this equation as: \( \frac{x^{2}}{9} + \frac{y^{2}}{36} + \frac{z^{2}}{25} = 1 \). This rearrangement helps us see the structure of the equation better.

Step 2: Recognize the Standard Form

The equation \( \frac{x^{2}}{9} + \frac{y^{2}}{36} + \frac{z^{2}}{25} = 1 \) is in the standard form of an ellipsoid equation, which is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \). Here, \( a = 3 \), \( b = 6 \), and \( c = 5 \). This standard form confirms that the surface we are working with is an ellipsoid.

Step 3: Use CAS to Plot the Surface

Next, we use a computer algebra system (CAS) to plot the surface represented by the equation \( \frac{x^{2}}{9} + \frac{y^{2}}{36} + \frac{z^{2}}{25} = 1 \). In the CAS, input the equation and adjust the view to clearly visualize the 3D shape of an ellipsoid.

Step 4: Identify and Label

After plotting, examine the graph to identify the geometric shape. Based on the previously determined equation form and the plot, label the graph as an ellipsoid. Note the orientation and axes lengths from the plot to confirm the type of quadric surface.

Key Concepts

Ellipsoid

An ellipsoid is a three-dimensional shape that resembles the form of a squished sphere. It is an important type of quadric surface characterized by its symmetrical shape. In mathematics, an ellipsoid has an equation in the form:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]This equation describes an ellipsoid centered at the origin, with axes aligned along the coordinate axes. Here, the constants \(a\), \(b\), and \(c\) are the semi-axes lengths of the ellipsoid.

• When \(a = b = c\), the ellipsoid is actually a sphere.
• When two semi-axes are equal and different from the third, it is sometimes called a spheroid.

In the exercise given, the ellipsoid is not a sphere since \(a\), \(b\), and \(c\) have different values (3, 6, and 5, respectively). This results in an ellipsoid elongated along certain axes based on the semi-axis lengths.
Ellipsoids are significant in various scientific fields such as physics and astronomy, often used to model planets and celestial bodies due to their rotation.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool that provides powerful means for performing symbolic mathematics and algebraic manipulations on a computer. It can handle a wide range of mathematical operations such as solving equations, plotting graphs, and manipulating algebraic expressions.
Using a CAS, one can input the equation of a quadric surface like an ellipsoid and easily visualize its form as a 3D plot. This is particularly useful when dealing with complex equations, enabling better understanding and identification of geometric shapes.

• CAS tools, such as Mathematica, Maple, or online platforms like Wolfram Alpha, can be used to plot surfaces.
• A CAS allows for interactive scaling, rotating, and zooming of graphs to see the quadric surface from different perspectives.

In the original task, a CAS is used to plot the ellipsoid, which helps to visually confirm the identity and properties of the surface based on its equation.

Standard Form

The standard form of a mathematical equation is its most simplified and recognizable version. For quadric surfaces, the standard form can greatly aid in identifying the type of surface at hand.
The ellipsoid standard form is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]This standard form makes it easy to see the relationship between the variables and the semi-axis lengths, providing immediate insight into how the surface is shaped.

• Equations in standard form should always equal a constant (e.g., 1 in the ellipsoid case).
• Rearranging the initial equation into its standard form is often a crucial step before interpreting or graphing.

In the given exercise, writing the equation in its standard form revealed that it was an ellipsoid. Understanding and using the standard form allows for clear identification, which is essential in fields such as geometry and calculus.