Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ 9 x^{2}+25 y^{2}+9 z^{2}=225 $$

Step-by-Step Solution

Answer

The equation represents an ellipsoid centered at the origin with axes lengths 5, 3, and 5.

Step 1: Identify the Type of Quadric Surface

The given equation is \( 9x^2 + 25y^2 + 9z^2 = 225 \). Each variable \( x, y, \) and \( z \) is squared, and their coefficients are positive, which suggests the graph is an ellipsoid.

Step 2: Rewrite the Equation in Standard Form

Divide each term by 225 to express the equation as \( \frac{x^2}{25} + \frac{y^2}{9} + \frac{z^2}{25} = 1 \). This is the standard form of an ellipsoid centered at the origin.

Step 3: Determine the Axes Lengths

From the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \), the radii along the \( x \), \( y \), and \( z \) axes are \( a = 5 \), \( b = 3 \), and \( c = 5 \) respectively.

Step 4: Sketch the Graph

Draw the 3D axes \( x, y, \) and \( z \). For the ellipsoid, the semi-major axes along \( x \) and \( z \) are 5 units, and the semi-minor axis along \( y \) is 3 units. Sketch an elongated shape stretched in the \( xz \)-plane with a smaller radius in the \( y \)-direction, centered at the origin.

Key Concepts

Quadric Surface

A quadric surface is a fundamental object in three-dimensional geometry. These surfaces are the graphical representations of second-degree equations in three variables. A classic example is an ellipsoid, like the one described in the given exercise. In general, quadric surfaces can be classified into several categories, such as ellipsoids, paraboloids, hyperboloids, and cones.

Here's what to remember about a quadric surface, especially an ellipsoid:

• It is defined by an equation of the form: \( ax^2 + by^2 + cz^2 = d \).
• For an ellipsoid, all the coefficients \( a, b, \text{and} c \) are positive, ensuring that the surface is closed and bounded.
• To identify the type of quadric surface from an equation, check the signs and coefficients of the squared terms.

By understanding these characteristics, you can classify any equation and predict the shape of its graph.

3D Graph

Working with graphs in three dimensions extends our understanding of geometric forms beyond the flat plane. A 3D graph represents shapes like spheres, cylinders, and ellipsoids in a space with width, depth, and height.

When sketching a 3D graph for an ellipsoid:

• Firstly, draw three perpendicular lines to represent the \( x \), \( y \), and \( z \) axes.
• The ellipsoid will appear as an elongated sphere stretched along certain axes depending on the standard form transformation.
• Plot the length of each axis based on the values derived from the standard form.

By visualizing these elements in three-space, you can better grasp the inherent symmetry and space-enclosing property of an ellipsoid. Practicing sketching these shapes helps in developing a spatial understanding that is crucial for further studies in three-dimensional geometry.

Standard Form

The standard form of a quadric equation simplifies analysis and sketching by providing clear information about the shape. For an ellipsoid, the standard form is vital since it directly shows the equation's radii along each axis.

Transforming an ellipsoid equation into standard form involves:

• Dividing each term by the constant term to normalize the equation to equal \( 1 \).
• Representing it in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \), where \( a, b, \) and \( c \) are the lengths of the semi-axes.
• The values \( a^2, b^2, \text{and} c^2 \) correspond to the squared denominators, giving the dimensions of the ellipsoid along the \( x, y, \text{and} z \) axes.

Understanding the standard form permits straightforward identification of the ellipsoid's dimensions and facilitates accurate graphing in three-dimensional space.