Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![**Find Distance and Equations in \( \mathbb{R}^3 \)**
**Write an equation for a sphere centered at the point (6, -4, -8) with radius 7.**
---
### Explanation
The task involves finding the equation of a sphere in three-dimensional space (\( \mathbb{R}^3 \)). The general equation for a sphere with center \((h, k, l)\) and radius \(r\) is given by:
\[
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
\]
In this case, the center of the sphere is \((6, -4, -8)\), and the radius is 7. Plugging these values into the equation, we get:
\[
(x - 6)^2 + (y + 4)^2 + (z + 8)^2 = 49
\]
This equation represents a sphere in \( \mathbb{R}^3 \) centered at \((6, -4, -8)\) with a radius of 7 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55a44e3d-6d6a-4ecd-9343-ab7bc7f808c7%2Fbe20f729-fa99-437c-b68f-c14ad22efa04%2F0hyenmh_processed.png&w=3840&q=75)
Transcribed Image Text:**Find Distance and Equations in \( \mathbb{R}^3 \)**
**Write an equation for a sphere centered at the point (6, -4, -8) with radius 7.**
---
### Explanation
The task involves finding the equation of a sphere in three-dimensional space (\( \mathbb{R}^3 \)). The general equation for a sphere with center \((h, k, l)\) and radius \(r\) is given by:
\[
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
\]
In this case, the center of the sphere is \((6, -4, -8)\), and the radius is 7. Plugging these values into the equation, we get:
\[
(x - 6)^2 + (y + 4)^2 + (z + 8)^2 = 49
\]
This equation represents a sphere in \( \mathbb{R}^3 \) centered at \((6, -4, -8)\) with a radius of 7 units.
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