Problem 19 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-44.\) ELLIPSOIDS $$4 x^{2}+9 y^{2}+4 z^{2}=36$$

Step-by-Step Solution

Answer

The ellipsoid extends 3 units in x- and z-directions, 2 units in y-direction.

Step 1: Identify the Standard Form of an Ellipsoid

The standard form for an ellipsoid is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\). Compare the given equation to this standard form to determine the values of \(a\), \(b\), and \(c\).

Step 2: Simplify the Given Equation

The given equation is \(4x^{2} + 9y^{2} + 4z^{2} = 36\). Divide both sides by 36 to simplify it.\[\frac{4x^2}{36} + \frac{9y^2}{36} + \frac{4z^2}{36} = \frac{36}{36}.\]

Step 3: Reduce Each Term

Simplify the fractions: \[\frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{9} = 1.\]This implies \(a^2 = 9\), \(b^2 = 4\), and \(c^2 = 9\).

Step 4: Find the Axes Lengths

Since \( a^2 = 9 \), \( b^2 = 4 \), and \( c^2 = 9 \), we have \( a = 3 \), \( b = 2 \), and \( c = 3 \). The axes lengths of the ellipsoid are 3, 2, and 3.

Step 5: Sketch the Ellipsoid

Draw a 3D coordinate system. The ellipsoid is centered at the origin \((0,0,0)\). Its principal axes are along the x, y, and z coordinates. It extends 3 units in the x-direction, 2 units in the y-direction, and 3 units in the z-direction. Sketch an oval-like shape that touches the principal points 3 units along the x and z axes, and 2 units along the y-axis.

Key Concepts

Standard Form of an Ellipsoid

An ellipsoid is a three-dimensional geometric surface, which is an extension of the idea of an ellipse into three dimensions. Understanding its standard form helps identify the shape's properties. The standard form of an ellipsoid is given by the equation: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1. \] In this equation, each term represents how much "space" the ellipsoid takes up along a particular axis. By comparing this standard form to the given equation, one can deduce the size and shape characteristics of a specific ellipsoid. It is essential to first convert an ellipsoid's equation to its standard form to identify these attributes, which involves ensuring the right-hand side of the equation is equal to 1. This conversion allows for easy interpretation of the ellipsoid's dimensions and orientation.

Axes Lengths

The axes lengths are the key to determining the size of an ellipsoid in three-dimensional space. In the standard form equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \), the values for \( a \), \( b \), and \( c \) represent the lengths of the semi-axes of the ellipsoid.

• \( a \): The length of the semi-axis along the x-direction.
• \( b \): The length of the semi-axis along the y-direction.
• \( c \): The length of the semi-axis along the z-direction.

For example, for an equation simplified to \( \frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{9} = 1 \), we find that \( a = 3 \), \( b = 2 \), and \( c = 3 \). Therefore, the ellipsoid extends 3 units along the x- and z-axes and 2 units along the y-axis. Understanding these lengths provides insight into the size and proportion of the ellipsoid in each directional space.

3D Sketching

Sketching a 3D geometric figure such as an ellipsoid can initially seem challenging, but following a structured approach simplifies the task. Begin by recognizing that the ellipsoid is centered at the origin in a coordinate system. The center point is \( (0, 0, 0) \).To sketch the ellipsoid:

• Draw a 3D coordinate system with x, y, and z axes.
• Identify and mark the points where the ellipsoid extends to its maximum along each semi-axis: 3 units along x and z, 2 units along y.
• Sketch an oval shape that smoothly connects these maximum points, ensuring symmetry about the center.

Visualizing it as an elongated sphere that "stretches out" in different directions can help. These simple steps lead to an accurate sketch, which can aid in understanding the ellipsoid's spatial form.

Coordinate System

The coordinate system is the mathematical 'stage' on which geometric figures like ellipsoids sit. Understanding the setup of a coordinate system allows for the precise placement and sketching of 3D objects like the ellipsoid seen here. In three-dimensional Cartesian systems, you have three axes:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically.
• The z-axis, which runs perpendicularly, coming out of or going into the page or screen.

Each point in this 3D space is defined by a set of coordinates \((x, y, z)\). The center of our ellipsoid, for instance, is at \((0, 0, 0)\), meaning it is directly at the origin where all three axes intersect. Understanding this system helps not only in sketching accurately, but also in interpreting and calculating distances, angles, and more related to 3D geometric shapes.