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Chapter 12: Problem 20

Write the equation of the indicated sphere. Center \((-2,1,-5)\). radius \(\sqrt{7}\)

Short Answer

Expert verified
The equation of the sphere is \( (x + 2)^2 + (y - 1)^2 + (z + 5)^2 = 7 \).

Step by step solution

01

Understand the sphere equation format

The general equation of a sphere with center \( (h, k, l) \) and radius \( r \) is \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \). Identifying the variables for our problem is essential for setting the equation.
02

Identify the values of center and radius

The center of this sphere is given as \( (-2, 1, -5) \) which means \( h = -2 \), \( k = 1 \), and \( l = -5 \). The radius is given as \( \sqrt{7} \), so \( r = \sqrt{7} \).
03

Substitute the values into the sphere equation

Using the values identified, substitute them into the sphere equation format: \( (x - (-2))^2 + (y - 1)^2 + (z - (-5))^2 = (\sqrt{7})^2 \).
04

Simplify the terms

Simplify the equation: \( (x + 2)^2 + (y - 1)^2 + (z + 5)^2 = 7 \). The problem is now fully expressed as a standard sphere equation with the given center and radius.

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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 component in understanding its geometry. Imagine a bubble floating in space; the center is the exact middle of this bubble. In mathematical terms, this center is defined by a point in three-dimensional space, typically labeled as \((h, k, l)\).
In our exercise, the center of the sphere is given as \((-2, 1, -5)\). Each coordinate鈥擻(-2\) for \(x\), \(1\) for \(y\), and \(-5\) for \(z\)鈥攑inpoints the sphere's location in space.
  • Why the center matters: It determines where the sphere is located. Without knowing the center, you can't accurately represent the sphere's position.
  • Recognizing center coordinates: These are simply the values in the format \((h, k, l)\), representing the middle point of the sphere.
Radius of a Sphere
The radius is another vital element in the sphere's geometry, telling us how big the sphere is. If the center is the starting dot in a circle, the radius is the distance from this dot to any point on the sphere鈥檚 surface. It's consistent in all directions, creating a perfectly round shape.
In our case, the radius is \(\sqrt{7}\), indicating the distance from the center, \((-2, 1, -5)\), to the surface.
  • Understanding the radius: Think of it as the measure of the sphere鈥檚 size. It鈥檚 always a positive value.
  • Representing the radius: Since the radius can appear as a square root, squaring it to get \(r^2\) is often necessary for equations, like converting \(\sqrt{7}\) to \(7\).
Standard Form of a Sphere Equation
The standard form of a sphere equation is essential for describing and graphing a sphere mathematically. It鈥檚 like having the sphere鈥檚 recipe, telling us how to draw it precisely in 3D space. The formula is:\[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\] where \((h, k, l)\) is the center, and \(r\) is the radius.
In the context of our exercise, substituting the given center, \((-2, 1, -5)\), and radius, \(\sqrt{7}\), into the formula yields:\[(x + 2)^2 + (y - 1)^2 + (z + 5)^2 = 7\]
  • Why use the standard form? It simplifies the process of determining a sphere's equation and helps in visualizing its properties.
  • Importance of substitution: Substituting the correct values directly into the formula ensures the accuracy of the sphere's equation representation.
  • Simplification: By organizing the equation in the standard form, mathematical operations and graphs become easier to handle and interpret.

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Most popular questions from this chapter

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