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Chapter 13: Problem 2

Describe the level surface \(F(x, y, z)=0\).\(F(x, y, z)=x^{2}+y^{2}+z^{2}-25\)

Short Answer

Expert verified
The level surface described by the function \(F(x, y, z)=x^{2}+y^{2}+z^{2}-25=0\) is a sphere centered at the origin with a radius of 5.

Step by step solution

01

Identify the Type of Level Surface

Looking at the equation \(x^{2}+y^{2}+z^{2}-25=0\), it can be identified as the general form of equation for a sphere \(x^{2}+y^{2}+z^{2}=r^{2}\). The equation can be rewritten as \(x^{2}+y^{2}+z^{2}=25\).
02

Determine the Radius

Now, the radius \(r\) of the sphere can be determined by taking the square root of the value on the right-hand side of the equation. So, the radius \(r\) is \( \sqrt{25} = 5.\)
03

Describe the Level Surface

Finally, with the type of level surface (a sphere) and its radius identified, the original equation \(F(x, y, z)=x^{2}+y^{2}+z^{2}-25=0\) can be described as a sphere in three dimensions centered at the origin (0, 0, 0) with a radius of 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Spherical Equations
Spherical equations describe the set of points in a space that form a sphere. The general equation for a sphere is \(x^2 + y^2 + z^2 = r^2\), where \(r\) is the radius of the sphere, and \((x, y, z)\) are the coordinates of any point on the sphere. In our exercise, the equation is given as \(x^2 + y^2 + z^2 - 25 = 0\). By adding 25 to both sides, it becomes \(x^2 + y^2 + z^2 = 25\). Here we clearly see it follows the general form of a sphere's equation, confirming it represents a sphere.
Basics of Three-Dimensional Geometry
Three-dimensional geometry explores shapes and figures in a space that has depth, width, and height. In this realm, objects like spheres, cubes, and cylinders exist with unique properties. For spheres, every point is equidistant from a central point. This symmetry makes spheres intriguing objects to study, especially in exercises like this where understanding the relationship between the mathematical equation and its geometric representation is key. Here, the equation denotes a sphere in 3D space, centered at the origin.
Describing Surfaces in Geometry
When describing geometric surfaces, we refer to the shapes formed by multiple points satisfying an equation. A level surface is such a set, defined by a constant value of a function in three dimensions. In the case of our sphere, the level surface \(F(x, y, z) = 0\) is described by the equation \(x^2 + y^2 + z^2 = 25\). This describes a spherical surface where every point on it is 5 units away from the origin, representing a perfect sphere.
Calculating the Radius of a Sphere
Calculating the radius in spherical equations is straightforward. From the equation \(x^2 + y^2 + z^2 = r^2\), the value \(r\) is the square root of the constant on the right side. In our specific problem, \(x^2 + y^2 + z^2 = 25\), so the radius \(r\) is \(\sqrt{25} = 5\). This calculation confirms that each point on the sphere is 5 units away from its center at the origin, ensuring clarity in the geometric interpretation of the equation.

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Most popular questions from this chapter

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