Problem 14 - Algebra and Trigonometry

Question

\(11-14\) . Find an equation of a sphere with the given radius \(r\) and center \(C .\) $$ r=\sqrt{11} ; \quad \quad \quad(-10,0,1) $$

Step-by-Step Solution

Answer

The equation of the sphere is \((x+10)^2 + y^2 + (z-1)^2 = 11\).

Step 1: Identify Properties of a Sphere

To find the equation of a sphere, we need the center and radius. The center of the sphere is given as \((-10,0,1)\) and the radius is given as \(\sqrt{11}\).

Step 2: Recall the Equation of a Sphere

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

Step 3: Substitute the Center Coordinates

Substitute \(h = -10\), \(k = 0\), and \(l = 1\) into the formula. This gives us:\[(x+10)^2 + y^2 + (z-1)^2 = r^2\].

Step 4: Substitute the Radius

Substitute the radius \(r = \sqrt{11}\) into the equation. Since the formula requires \(r^2\), calculate \((\sqrt{11})^2 = 11\). This changes the equation to:\[(x+10)^2 + y^2 + (z-1)^2 = 11\].

Step 5: Final Equation

Now we have the final equation of the sphere:\[(x+10)^2 + y^2 + (z-1)^2 = 11\].

Key Concepts

geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. Spheres are perfectly symmetrical objects in three dimensions, and they play an important role in geometric studies. Understanding the geometry of spheres is essential for identifying their properties and creating their mathematical descriptions.
To find the equation of a sphere, you need two crucial pieces of information:

• The center of the sphere, which is a point in space
• The radius, which is the distance from the center to any point on the surface

The symmetry and uniformity of spheres mean that the same distance can be measured in any direction from the center. This characteristic leads to the equation often associated with spheres in three-dimensional geometry.In this exercise, we're given the center ofthe sphere at o(x= -10, y=0, z=1)o and the radius, o\(\sqrt{11}\). Using these, the equation of the sphere is derived as explained in the step-by-step solution.

3D coordinate system

The 3D coordinate system is a framework used to describe the position of points in three-dimensional space. It consists of three axes - the x-axis, y-axis, and z-axis - that intersect at a point called the origin, forming a right angle with each other. Each point in this space is represented as a triplet o\((x, y, z)\), where o(x, y, z)o are the coordinates determining its position along the respective axes.
The 3D coordinate system is crucial when working with spheres, as it helps to precisely locate the sphere's center and describe its equation. In this context:

• The x-coordinate represents the horizontal distance from the origin.
• The y-coordinate reflects the vertical distance from the origin.
• The z-coordinate shows the depth from the origin signifying a point's position in the third dimension.

In the exercise, our center o\((-10, 0, 1)\) is defined within this 3D coordinate framework. This allows us to use the standard sphere equation to calculate distances from this center in all directions.

radius and center of sphere

The radius and center are fundamental concepts when working with the geometry of spheres. The radius is a constant distance from the center to any point on the sphere's surface. In contrast, the center is a fixed point inside the sphere from which every point on the surface is equidistant. Knowing these allows us to translate a sphere's geometric properties into mathematical form.
For spheres, the typical equation o\((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\) displays this relationship, where o\((h, k, l)\) is the center, and o\(r\) is the radius. By substituting the center and radius into this equation, you can describe any sphere in 3D space.Looking back at the original exercise, the center o\((-10, 0, 1)\) and radius o\(\sqrt{11}\) are vital for forming the specific sphere equation, o\((x+10)^2 + y^2 + (z-1)^2 = 11\). This equation succinctly captures all the essential properties of the sphere described.