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Chapter 11: Problem 47

Find the centers and radii of the spheres. $$(x+2)^{2}+y^{2}+(z-2)^{2}=8$$

Short Answer

Expert verified
The center of the sphere is (-2, 0, 2) with a radius of 2鈭�2.

Step by step solution

01

Identify the Sphere Equation

The given equation is \((x+2)^2 + y^2 + (z-2)^2 = 8\). This equation is in the standard form of a sphere's equation, which is \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\).
02

Extract the Center Coordinates

Match the terms in the given sphere's equation with the standard form. The equation \((x+2)^2\) indicates that \(h = -2\), the term \(y^2\) shows \(k = 0\), and \((z-2)^2\) means \(l = 2\). Therefore, the center of the sphere is at \((-2, 0, 2)\).
03

Determine the Radius

The equation of the sphere is set equal to \(8\). The term on the right side, \(r^2 = 8\), means the radius \(r = \sqrt{8}\). Simplify this to get \(r = 2\sqrt{2}\).

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Key Concepts

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

Center of a Sphere
The center of a sphere is a crucial concept when dealing with sphere equations. It represents the point in space from which all points on the surface of the sphere are equidistant. To find the center from an equation given in standard form, look at the expression \[(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\] Here, the center
  • is denoted by the point \((h, k, l)\).

In the specific problem equation \((x+2)^2 + y^2 + (z-2)^2 = 8\),
  • compare each part of the squared terms with the standard form:
  • \(h = -2\), corresponding to \((x+2)^2\),
  • \(k = 0\), from \(y^2\) where the absence of addition or subtraction implies \(k = 0\),
  • and \(l = 2\), reflected in \((z-2)^2\).
This means that the center of the sphere is located at \((-2, 0, 2)\). This point serves as the anchor for the entire sphere's geometry.
Radius of a Sphere
The radius of a sphere is the distance from the center to any point on the surface. Understanding and calculating the radius is essential since it defines the sphere's size. It is derived from the equation's right side in the standard form \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), where \(r^2\) stands for the square of the radius.
Using the equation \((x+2)^2 + y^2 + (z-2)^2 = 8\),
  • we identify \(r^2 = 8\).
  • To find the actual radius, take the square root of this value: \(r = \sqrt{8}\).
  • This can be simplified to \(r = 2\sqrt{2}\).
So the radius of the sphere we are dealing with is \(2\sqrt{2}\). Knowing the radius allows you to understand how large or small the sphere is compared to other spheres.
Standard Form of a Sphere
The standard form of a sphere equation is a fundamental aspect of geometry that allows us to easily determine the sphere's center and radius. This form, written as \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\),permits a straightforward identification of the center and radius.
  • Each squared term \((x-h)^2\), \((y-k)^2\), and \((z-l)^2\)helps you identify the sphere's center coordinates \(h\), \(k\), and \(l\).
  • The right-hand side, \(r^2\), gives essential information about the sphere's radius.

Working with the sphere's equation: * Enables problem-solving in three-dimensional geometry. * Assists in visualizing and calculating distances in space.Understanding this standard form boosts confidence with varied applications ranging from geometry to physics and engineering.

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