Problem 18 - University Calculus: Early Transcendentals

Question

Sketch the surfaces ELLIPSOIDS $$4 x^{2}+4 y^{2}+z^{2}=16$$

Step-by-Step Solution

Answer

Ellipsoid centered at (0, 0, 0), semi-axes (2, 2, 4), elongated along z-axis.

Step 1: Identify the standard form

First, let's identify the standard form of an ellipsoid equation. The standard form is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\). In order to compare, we should rewrite the given equation \(4x^{2} + 4y^{2} + z^{2} = 16\) to this standard form.

Step 2: Rearrange the equation

Divide every term in the equation \(4x^{2} + 4y^{2} + z^{2} = 16\) by 16 to rewrite it as \(\frac{4x^2}{16} + \frac{4y^2}{16} + \frac{z^2}{16} = 1\). This simplifies to \(\frac{x^{2}}{4} + \frac{y^{2}}{4} + \frac{z^{2}}{16} = 1\).

Step 3: Interpret the parameters

The equation \(\frac{x^{2}}{4} + \frac{y^{2}}{4} + \frac{z^{2}}{16} = 1\) indicates the axes lengths of the ellipsoid. Here, \(a^{2} = 4\), \(b^{2} = 4\), \(c^{2} = 16\), which gives the principal semi-axes lengths as \(a = 2\), \(b = 2\), \(c = 4\). This means the ellipsoid is stretched along the z-axis.

Step 4: Sketch the ellipsoid

Draw a three-dimensional coordinate system. Since \(a = 2\) and \(b = 2\), the base of the ellipsoid in the xy-plane is a circle with radius 2. The height along the z-axis is 4, so the ellipsoid is elongated along the z-axis. This ellipsoid is centered at the origin (0, 0, 0) in three-dimensional space.

Key Concepts

Standard Form of Ellipsoids

In geometry, an ellipsoid is a three-dimensional shape that generalizes the concept of a two-dimensional ellipse. The **standard form** of the equation of an ellipsoid is:
\[\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1\]Here,

• \(a\), \(b\), and \(c\) denote the lengths of the semi-principal axes of the ellipsoid
• The denominator of each term represents the square of these axes lengths
• The equation is normalized such that it equals 1

To convert any given ellipsoid equation into this standard form, all terms can be equalized to one side of the equation, forming a ratio set equal to 1. This illustrates how each variable is squared and divided by a squared constant, allowing us to easily compare it to the standard form. If any real-world application requires an understanding of how different parts coordinate spatially, translating the equation into this standard format is the way to gain clarity.

Axes Lengths

The term **axes lengths** in relation to ellipsoids refers to the important measurements that describe the size and orientation of the ellipsoid's axes. When the equation \[ \frac{x^{2}}{4} + \frac{y^{2}}{4} + \frac{z^{2}}{16} = 1 \] is observed:

• The values 4, 4, and 16 in the denominators correspond to \(a^2, b^2,\) and \(c^2\)
• The principal semi-axes are determined as \(a = 2, b = 2,\) and \(c = 4\)

The lengths of these semi-axes determine the shape of the ellipsoid. In this case, since the length along the z-axis is greater (\(c = 4\)), the ellipsoid is elongated vertically compared to the horizontally equal lengths of axes \(a\) and \(b\). Recognizing these dimensions is crucial for understanding the spatial properties of the ellipsoid and can help in visualizing its shape relative to the three-dimensional coordinate grid.

Three-Dimensional Coordinates

**Three-dimensional coordinates** are vital to understanding and visualizing ellipsoids in space. In the coordinate system:

• The z-axis runs vertically
• The x and y-axes run horizontally, forming the familiar plane

For ellipsoids, such as the one with the equation \( \frac{x^{2}}{4} + \frac{y^{2}}{4} + \frac{z^{2}}{16} = 1 \), it is centered at the origin point (0, 0, 0) within these coordinates. The axes lengths influence how stretched or compressed the ellipsoid looks in each direction. This ellipsoid features a base circle on the xy-plane due to equal axes of \(a\) and \(b\), with its height extending further along the z-axis. Thus, visualizing the ellipsoid requires understanding that it occupies space around this central origin, stretching dynamically according to the prescribed geometric axes lengths. Comprehending this can be particularly beneficial if you're engaged in sciences or disciplines where three-dimensional modeling is necessary.