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Chapter 11: Problem 52

Find equations for the spheres whose centers and radii are given. Center (0,-1,5) Radius 2

Short Answer

Expert verified
The equation of the sphere is \\(x^2 + (y+1)^2 + (z-5)^2 = 4\\).

Step by step solution

01

Understand the Equation of a Sphere

The formula for the equation of a sphere with center \(a, b, c\) and radius \ R \ is given by: \((x-a)^2 + (y-b)^2 + (z-c)^2 = R^2\).
02

Substitute the Center Coordinates

Given the center of the sphere is (0, -1, 5), substitute these into the equation: \((x-0)^2 + (y+1)^2 + (z-5)^2 = R^2\).
03

Substitute the Radius

The radius of the sphere is given as 2. Substitute this value into the equation: \((x-0)^2 + (y+1)^2 + (z-5)^2 = 2^2\).
04

Simplify the Equation

Simplify the equation to get the standard form of the sphere: \(x^2 + (y+1)^2 + (z-5)^2 = 4\).

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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 starting point when defining the geometric shape. Imagine a point in the middle of your sphere. This point is referred to as the "center" and serves as the point from which every spot on the surface of the sphere is equidistant. In mathematical terms, the center of a sphere is often denoted by coordinates
  • Typically, these coordinates are labeled as \((a, b, c)\), representing the center's location in three-dimensional space.
  • For example, with a center at \((0, -1, 5)\), each number corresponds to the \(x\), \(y\), and \(z\) positions in space.
  • To find the center of a sphere, look for these coordinates in the provided data or description of your sphere.
Radius of a Sphere
The radius of a sphere reaches from the center to any point on its surface, always maintaining the same length. It's a constant measure for any sphere.
  • In sphere equations, the radius is typically represented by the variable \( R \).
  • Given values, like a radius of 2 units, provide the fixed distance as seen in our example.
  • The radius is crucial because it helps form the equation of the sphere and gives insight into the sphere's size.
To understand it visually, imagine drawing a line segment from the center straight to the sphere's outer edge. This line鈥檚 span is the radius. Use this critical piece of information to complete the sphere's equation effectively.
Standard Form of a Sphere Equation
The standard equation for a sphere brings everything together鈥攖he center and the radius.This equation is written generally as: \[(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2\]
  • The variables \(x, y, z\) describe any point on the sphere's surface.
  • The terms \(a, b, c\) are the coordinates of the center.
  • \(R\) is the radius, which gets squared in the equation.
For example, for our sphere with center \((0, -1, 5)\) and radius 2, substitute these values:\[(x-0)^2 + (y+1)^2 + (z-5)^2 = 2^2\]Simplifying gives\[x^2 + (y+1)^2 + (z-5)^2 = 4\]This formula creates a clear, mathematical representation of any sphere given its central point and size. It's a staple in geometry for understanding spherical shapes in a straightforward manner.

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Most popular questions from this chapter

a. Express the area \(A\) of the cross-section cut from the ellipsoid $$ x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}=1 $$ by the plane \(z=c\) as a function of \(c .\) (The area of an ellipse with semiaxes \(a \text { and } b \text { is } \pi a b .)\) b. Use slices perpendicular to the \(z\) -axis to find the volume of the ellipsoid in part (a). c. Now find the volume of the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$ Does your formula give the volume of a sphere of radius \(a\) if \(a=b=c ?\)

Find parametrizations for the lines in which the planes. $$x-2 y+4 z=2, \quad x+y-2 z=5$$

Find the acute angles between the planes in Exercises \(49-52\) to the nearest hundredth of a radian. $$2 x+2 y+2 z=3, \quad 2 x-2 y-z=5$$

Find parametrizations for the lines in which the planes. $$3 x-6 y-2 z=3, \quad 2 x+y-2 z=2$$

In Exercises \(35-38,\) find a. the direction of \(\overrightarrow{P_{1} P_{2}}\) and b. the midpoint of line segment \(P_{1} P_{2}\) $$P_{1}(0,0,0) \quad P_{2}(2,-2,-2)$$
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