Problem 41 - Thomas Calculus

Question

Find the centers and radii of the spheres in Exercises \(41-44\) $$ (x+2)^{2}+y^{2}+(z-2)^{2}=8 $$

Step-by-Step Solution

Answer

The center is (-2, 0, 2) and the radius is \(2\sqrt{2}\).

Step 1: Identify the equation

The given equation of the sphere is \[(x+2)^2 + y^2 + (z-2)^2 = 8\]This is in the standard form of a sphere \[(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\]where \( (h, k, l) \) represents the center and \( r \) is the radius.

Step 2: Extract the center from the equation

To find the center of the sphere, compare the given equation with the standard form. From \[(x+2)^2 + y^2 + (z-2)^2 = 8\]we identify that \( h = -2, k = 0, \) and \( l = 2 \).Thus, the center of the sphere is \( (-2, 0, 2) \).

Step 3: Determine the radius

The right side of the equation \[(x+2)^2 + y^2 + (z-2)^2 = 8\]represents \( r^2 \). Hence, we have \( r^2 = 8 \). Therefore, the radius \( r \) is given by the square root of 8, which is \( \sqrt{8} = 2\sqrt{2} \).

Key Concepts

Center of Sphere

The center of a sphere is a fundamental concept that plays a critical role in understanding geometric equations of spheres. Essentially, the center of a sphere is the point from which every point on the surface of the sphere is equidistant. In mathematical terms, this point is represented by the coordinates

Standard Sphere Equation

The standard form of a sphere's equation is \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), where

• \((h, k, l)\) are the coordinates of the center of the sphere.
• \(r\) is the radius of the sphere.

By inspecting the standard form and matching it with our exercise equation \((x+2)^2 + y^2 + (z-2)^2 = 8\), we can identify the center coordinates as \((-2, 0, 2)\). This is because each part of the squared term corresponds to its respective coordinate shifted by a number to reflect the sphere's center. The negative and positive values indicate how these coordinates must be adjusted in the graph.

Radius of Sphere

The radius of a sphere is the distance from the center point to any point on its surface. Understanding the concept of the radius is crucial as it helps determine how large or small the sphere is in space.
In the context of geometric equations, the radius is represented as \(r\).

Calculating the Radius

To find the radius from the equation \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), we focus on the value of \(r^2\). In our specific example, \((x+2)^2 + y^2 + (z-2)^2 = 8\), the term \(8\) represents \(r^2\). The next step is to calculate the square root of this number, which gives us the actual radius:
\(r = \sqrt{8} = 2\sqrt{2}\). Knowing the radius allows for deeper geometric understanding, allowing for calculations of volume and surface area, too.

Geometric Equations

Geometric equations provide a powerful way to describe and analyze all shapes and figures through mathematical expressions. For spheres, these equations allow us to model a perfectly round three-dimensional object using algebra.

Building and Understanding Equations

Understanding the standard sphere equation is essential: \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\).This represents a space in which a perfect sphere exists, defined by:

• Its center, determined by \((h, k, l)\).
• Its radius, defined by the square root of the equation's right-hand side.

Using this formula, one can convert spatial properties into understandable algebraic expressions. At the same time, it enables practical usage like graph plotting and ensuring precision in 3D spatial design. Mastery of geometric equations facilitates not only solving textbook problems but enhancing spatial reasoning and logical thinking.