Problem 43 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

Center: \((\sqrt{2}, \sqrt{2}, -\sqrt{2})\), Radius: \(\sqrt{2}\).

Step 1: Identify the General Equation

The general equation of a sphere is \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), where \((h, k, l)\) is the center and \(r\) is the radius.

Step 2: Compare with General Form

The given equation is \((x-\sqrt{2})^2 + (y-\sqrt{2})^2 + (z+\sqrt{2})^2 = 2\). By comparing this with the general form \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), we can identify the center and the radius.

Step 3: Determine the Center of the Sphere

From \((x-\sqrt{2})\), we get \(h = \sqrt{2}\). From \((y-\sqrt{2})\), we get \(k = \sqrt{2}\). From \((z+\sqrt{2})\), note it can be written as \((z-(-\sqrt{2}))\), therefore \(l = -\sqrt{2}\). So, the center is \((\sqrt{2}, \sqrt{2}, -\sqrt{2})\).

Step 4: Determine the Radius of the Sphere

The right side of the equation \(2\) is \(r^2\). Therefore, the radius \(r\) is found by taking the square root of \(2\). So \(r = \sqrt{2}\).

Key Concepts

Understanding Sphere Geometry

In geometry, a sphere is a three-dimensional object where every point on its surface is equidistant from a central point, known as the center. This centrality gives the sphere its perfectly symmetrical shape. It's important to remember the equation of a sphere:

• General form: \[ (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \]
• The variables \((h, k, l)\) represent the coordinates of the center of the sphere.
• \( r \)is the radius, the distance from the center to any point on the sphere.

This equation helps you determine the position and size of the sphere in any three-dimensional coordinate system. Recognizing this form enables us to identify the center and the radius from any sphere equation given in this standard format.

Steps to Calculate the Radius

Calculating the radius of a sphere from its equation is a straightforward process. The key is to recognize which part of the equation represents \(r^2\). Begin by looking at the equation's right side:

• If the equation is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), then \(r^2\) is simply the constant on the right side.
• Take the square root of this constant to find the radius, \( r \).

In our specific example, the right side is equal to 2. Thus, the radius, \( r, \) is \( \sqrt{2} \).By finding the radius, you understand how big the sphere is since it denotes the full extent of its surface from the center.

Methods for Center Determination

Determining the center of a sphere involves examining the transformations in the sphere equation's brackets. For each variable (x, y, and z), identify the terms within the brackets:

• For \((x - h)\), \(h\) is extracted as it signifies the x-coordinate of the center.
• For \((y - k)\), \(k\) represents the y-coordinate.
• For \((z - l)\), \(l\) indicates the z-coordinate.

In the example provided, the expression \((x - \sqrt{2})\) yields \(h = \sqrt{2}\), and similarly with y. For z, since it's in the form \((z + \sqrt{2})\), rewrite it as \( (z - (- \sqrt{2})) \).This clarifies that \(l = - \sqrt{2} \).Thus, the full center coordinates are \((\sqrt{2}, \sqrt{2}, -\sqrt{2})\), giving you the exact point everything rotates around in the sphere.