Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ 4 x^{2}+9 y^{2}+49 z^{2}=1764 $$

Step-by-Step Solution

Answer

The graph is an ellipsoid centered at the origin with semi-axes lengths 21, 14, and 6.

Step 1: Identify the Type of Equation

First, we recognize the form of the given equation \(4x^{2} + 9y^{2} + 49z^{2} = 1764\). It represents a quadratic surface since each variable \(x\), \(y\), and \(z\) is squared.

Step 2: Rewrite in Standard Form

To better understand the structure, we rewrite this equation in its standard form. Divide every term by 1764: \[\frac{x^2}{441} + \frac{y^2}{196} + \frac{z^2}{36} = 1.\]

Step 3: Identify the Type of Quadric Surface

The equation \(\frac{x^2}{441} + \frac{y^2}{196} + \frac{z^2}{36} = 1\) is in the standard form for an ellipsoid: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\).

Step 4: Identify the Axes Lengths

Identify the semi-axes lengths from the standard form: \(a = 21\) (since \(21^2 = 441\)), \(b = 14\) (since \(14^2 = 196\)), and \(c = 6\) (since \(6^2 = 36\)).

Step 5: Sketch the Ellipsoid

Draw the ellipsoid centered at the origin \((0, 0, 0)\) in three-space, with axes: the x-axis extending to \(\pm 21\), the y-axis to \(\pm 14\), and the z-axis to \(\pm 6\).

Key Concepts

Quadric Surfaces

In the realm of 3D geometry, many surfaces can be described by algebraic equations involving squared terms. These are known as quadric surfaces. The equation we started with, \(4x^{2} + 9y^{2} + 49z^{2} = 1764\), is an example of such a surface.

Quadric surfaces come in several types, each with its own distinctive shape:

• Ellipsoids: These are egg-shaped or sphere-like surfaces.
• Hyperboloids: These can be either one-sheet or two-sheet, with saddle-like or two distinct caps.
• Cones: Shapes that converge at a point or an axis.
• Elliptic Paraboloids: Shapes similar to an elongated parabola.

Recognizing a quadric surface typically starts with observing that each term in an equation is a squared variable. This alone signals the presence of a possible quadric surface.

Ellipsoid

When exploring 3D shapes like quadric surfaces, the ellipsoid stands out as an important form. It resembles a distorted sphere and represents a variety of forms from the ordinary ball to more stretched or flattened spheroids.

The standard equation for an ellipsoid in three-space is: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]
Each term involves both the square of a coordinate and a constant in the denominator, which are crucial in defining the ellipsoid's dimensions.

Unlike perfect spheres, ellipsoids encompass various dimensions along the three axes, making them versatile in describing many real-world objects from planets to basketballs. They are common not just in mathematics but in physics and engineering too.

Equation of Three-Space

Three-space, or 3D space, is a mathematical concept offering a framework for visualizing equations beyond two dimensions. This concept allows us to imagine shapes like ellipsoids, hyperboloids, and even paraboloids.

When an equation includes different variables like x, y, and z, each squared, it represents a shape existing in this three-dimensional world. For example, the equation \(\frac{x^2}{441} + \frac{y^2}{196} + \frac{z^2}{36} = 1\) describes a closed surface in this space.

Understanding three-space is key to mastering concepts in geometry, physics, and even modern graphics technology. It allows for modeling complex systems and shapes, giving us a deeper insight into the physical world.

Axes Lengths

The lengths of the axes in an ellipsoid play a crucial role in defining its shape. For the equation \(\frac{x^2}{441} + \frac{y^2}{196} + \frac{z^2}{36} = 1\), each denominator represents the square of the axis length along the respective coordinate axes.

To determine the actual lengths:

• x-axis: The semi-major axis is \(a = 21\), since \(21^2 = 441\).
• y-axis: The semi-major axis is \(b = 14\), since \(14^2 = 196\).
• z-axis: The semi-major axis is \(c = 6\), since \(6^2 = 36\).

These values give the ellipsoid its three principal radii, determining its stretch and form in the 3D space. The axes lengths help to visualize the orientation and scale of the ellipsoid within its coordinate system.