Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The graph is a double cone symmetrical about the x-axis.

Step 1: Identify the Equation Type

The given equation is \[-x^2 + y^2 + z^2 = 0\]. This equation is of the form \[-x^2 + y^2 + z^2 = 0\]. This resembles the equation of a cone in three-dimensional space, where \(x^2 = y^2 + z^2\).

Step 2: Rewrite the Equation

Rearrange the equation to isolate \(x^2\) on one side:\[x^2 = y^2 + z^2\].This confirms that the surface is a double cone with its vertex at the origin, opening in the positive and negative x-directions.

Step 3: Analyze the Shape

The equation \(x^2 = y^2 + z^2\) represents a double cone. If we slice the graph with a plane containing the x-axis (i.e., the yz-plane), the sections will be a pair of lines crossing at the origin. If we fix x to any positive or negative constant, the resulting equation is a circle.

Step 4: Sketch the Graph

Begin by sketching the yz-plane with the cone opening symmetrically around the x-axis, since the cone extends indefinitely on both sides of the origin along the x-axis. The intersection with any plane parallel to the yz-plane results in a circle, reflecting the equation \(x^2 = y^2 + z^2\).

Step 5: Verify by Cross Sections

For constant values of \(x\), say \(x = c\), the equation becomes \[c^2 = y^2 + z^2\].This is a circle in the yz-plane with center at the origin and radius \(c\). This confirms the graph is a cone with circular cross-sections.

Key Concepts

Double Cone

A double cone is a fascinating three-dimensional shape. Imagine two ice cream cones connected at their tips, sharing the same axis.

• Each cone extends infinitely in opposite directions from the vertex.
• It is symmetrical around the axis that passes through its vertex.
• A plane intersecting the cone through its vertex produces two lines.

Double cones in equations, like the one given \[x^2 = y^2 + z^2\]represent shapes that extend infinitely along one axis, forming a circular cross-section in any perpendicular plane. These cones span two regions: one in the direction of positive x-values and one in the direction of negative x-values.

Three-Dimensional Space

Three-dimensional space is the environment where these equations come to life. It's often referred to as 3D space and can be visualized as a vast grid.

• It includes three axes: x, y, and z, each representing a different dimension.
• Objects like cones can have depth, width, and height in this space.
• When graphing, it's important to understand the coordination between these three axes.

In this context, the given equation describes a shape that lives in this space, where movements along any axis affect the shape's appearance and position. Cones in this space are rich in symmetry and depth, having sections that reveal different shapes when intersected by planes.

Equation of a Cone

The equation of a cone is essential for understanding its geometry.

• A standard cone equation can be written as \[x^2 = y^2 + z^2\].
• This represents a perfect cone with its axis along the x-line.
• The vertex is typically placed at the origin, allowing symmetric extension.

With the equation \[-x^2 + y^2 + z^2 = 0\], you can see the structure by rearranging terms to identify the double cone characteristics. This equation highlights how different values for x produce circular cross-sections within the plane parallel to the yz-plane, portraying how the shape behaves dynamically in 3D space.

Cartesian Coordinates

Cartesian coordinates are the backbone for 3D graphing. They help us place every point in space accurately.

• Defined by three values (x, y, and z) for three-dimensional space.
• These coordinates help in visualizing the position and relation of points and shapes.
• Equations convert easily to visual representations when plotted on Cartesian axes.

Through Cartesian coordinates, placing and interpreting shapes like a double cone becomes manageable. By reading coordinates, one can understand the specific sections of a cone, like circles or lines, determined by intersection with planes, such as the yz-plane. This system simplifies complex problems by aligning them with a familiar coordinate structure, aiding in insight and problem-solving.