Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ 2 x^{2}-16 z^{2}=0 $$

Step-by-Step Solution

Answer

The equation represents a double cone centered on the origin with an axis along the y-axis.

Step 1: Identify the Equation Type

The given equation is \(2x^2 - 16z^2 = 0\). This equation represents a cone in three-dimensional space. The terms \(x^2\) and \(z^2\) having opposite signs and the absence of a \(y\) term indicates it relates to a conical structure with an axis along the axis not included \(x=0, z=0\).

Step 2: Simplify the Equation

Start by simplifying the equation: \(2x^2 - 16z^2 = 0\).1. Factor out the common factor on both sides: \(x^2 - 8z^2 = 0\)2. Rearrange into a factored form: \((x - 0) = \pm \sqrt{8}z \)This represents a double cone opening along the y-axis.

Step 3: Sketch the Graph

The form \((x - 0) = \pm \sqrt{8}z \) represents a double cone that opens up and downward along the \(y\)-axis in three-dimensional space. To sketch it:1. Consider \(x\)-\(z\) cross-sections: for every constant \(y=k\), the equation represents lines intersecting at the origin, forming V-shapes.2. As \(y\) moves along the \(y\)-axis, these V-shapes expand.3. Combine all cross-sections to visualize the cone's 3D shape.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. There are four main types of conic sections: circles, ellipses, parabolas, and hyperbolas. These shapes are fundamental in mathematics because they naturally arise in many practical situations, such as planetary orbits and satellite pathways.
A conic section can be described by a polynomial equation of the second degree in two variables. Depending on the signs and coefficients of these variables, the specific conic type is determined.

• When the plane is perpendicular to the cone's axis and intersects both nappes, the conic is a hyperbola.
• In the case when the plane is parallel to the axis of the cone, a parabola is formed.
• If the plane cuts parallel to the cone's base, a circle or an ellipse is formed, depending on the angle.

Understanding these basic conic forms helps learn about their 3D counterparts, relevant in the context of 3D geometry.

Graphing Equations

Graphing equations in a three-dimensional context involves understanding how equations define shapes through variables and constants across different dimensions. To graph equations effectively, particularly in 3D space, follow a simple yet systematic approach:

• Identify the type of geometric shape the equation represents by analyzing terms and their coefficients. For example, terms that negate one another, like in the equation for a cone, hint towards hyperbolic or conic structures.
• Simplify the equation if possible. For instance, simplifying \(2x^2 - 16z^2 = 0\) to \(x^2 - 8z^2 = 0\) makes it clearer and easier to understand the geometric implications.
• Determine the axis along which the shape aligns. In the simplified cone equation, the lack of a \(y\) term immediately suggests that the cone symmetrically opens along the \(y\)-axis.

Practicing these steps not only aids in sketching but also in visualizing complex geometrical forms that arise from algebraic equations.

Three-Dimensional Space

Three-dimensional space, often referred to as 3D space, is a geometric model of the physical universe in which we live. It incorporates three dimensions: length, width, and height (often denoted as \(x\), \(y\), and \(z\)). Each point in this space can be identified by three coordinates, making it an extension of the traditional two-dimensional coordinate system.
When dealing with equations and shapes in 3D space, the challenge often lies in visualizing these structures and understanding their interactions across different axes. For example, consider a double cone like the one described in the exercise. This geometric shape is centered around its axis, expanding and shrinking as you move along the axis not included in the equation.

• The given cone equation \((x - 0) = \pm \sqrt{8}z\) depicts a cone symmetric around the \(y\)-axis.
• It is crucial to recognize the symmetry and orientation of shapes in 3D space to sketch and understand them accurately.
• Visualizing 3D objects often involves looking at 2D cross-sections, such as parallel planes to the axes, which help indicate how the object expands or shrinks.

Mastery of these concepts enables a better understanding of the physical world and geometrical relationships.