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
Can anyone please help me to solve this problem? I am stuck! Please help me!
![**Problem Statement**
Find the equation of the sphere if one of its diameters has endpoints \((6, -7, 8)\) and \((8, -3, 19)\).
**Solution Steps**
1. **Find the Center of the Sphere:**
The center of the sphere is the midpoint of the diameter. Use the midpoint formula:
\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]
For the given points \((6, -7, 8)\) and \((8, -3, 19)\), calculate:
\[
\left( \frac{6 + 8}{2}, \frac{-7 + (-3)}{2}, \frac{8 + 19}{2} \right)
\]
2. **Calculate the Radius:**
The radius is half the distance between the endpoints of the diameter. Use the distance formula:
\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
For our points \((6, -7, 8)\) and \((8, -3, 19)\), calculate:
\[
\sqrt{(8-6)^2 + (-3 - (-7))^2 + (19 - 8)^2}
\]
Then divide by 2 to find the radius.
3. **Formulate the Equation of the Sphere:**
Use the standard equation of a sphere:
\[
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
\]
Substitute \((h, k, l)\) with the center found in step 1 and \(r\) with the radius calculated in step 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2F67d15ef0-13a1-44df-a956-7cdcb217f096%2Fkkthz9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the equation of the sphere if one of its diameters has endpoints \((6, -7, 8)\) and \((8, -3, 19)\).
**Solution Steps**
1. **Find the Center of the Sphere:**
The center of the sphere is the midpoint of the diameter. Use the midpoint formula:
\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]
For the given points \((6, -7, 8)\) and \((8, -3, 19)\), calculate:
\[
\left( \frac{6 + 8}{2}, \frac{-7 + (-3)}{2}, \frac{8 + 19}{2} \right)
\]
2. **Calculate the Radius:**
The radius is half the distance between the endpoints of the diameter. Use the distance formula:
\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
For our points \((6, -7, 8)\) and \((8, -3, 19)\), calculate:
\[
\sqrt{(8-6)^2 + (-3 - (-7))^2 + (19 - 8)^2}
\]
Then divide by 2 to find the radius.
3. **Formulate the Equation of the Sphere:**
Use the standard equation of a sphere:
\[
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
\]
Substitute \((h, k, l)\) with the center found in step 1 and \(r\) with the radius calculated in step 2.
![**Problem Statement:**
Find the equation of the sphere if one of its diameters has endpoints \((6, 7, 8)\) and \((8, -3, 19)\).
**Solution Approach:**
To find the equation of the sphere, follow these steps:
1. **Find the Center of the Sphere:**
The center of the sphere is the midpoint of the diameter. Use the midpoint formula:
\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]
2. **Calculate the Midpoint:**
Substitute the given points \((6, 7, 8)\) and \((8, -3, 19)\):
\[
\left( \frac{6 + 8}{2}, \frac{7 + (-3)}{2}, \frac{8 + 19}{2} \right) = (7, 2, 13.5)
\]
3. **Determine the Radius:**
The radius is half the distance between the two endpoints. Use the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
4. **Calculate the Distance:**
\[
\sqrt{(8 - 6)^2 + (-3 - 7)^2 + (19 - 8)^2} = \sqrt{2^2 + (-10)^2 + 11^2} = \sqrt{4 + 100 + 121} = \sqrt{225} = 15
\]
The radius \(r\) is half of 15, so \(r = 7.5\).
5. **Equation of the Sphere:**
Use the standard equation of a sphere:
\[
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
\]
Substitute \(h = 7\), \(k = 2\), \(l = 13.5\), and \(r = 7.5\):
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2F67d15ef0-13a1-44df-a956-7cdcb217f096%2Fw9p831_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the equation of the sphere if one of its diameters has endpoints \((6, 7, 8)\) and \((8, -3, 19)\).
**Solution Approach:**
To find the equation of the sphere, follow these steps:
1. **Find the Center of the Sphere:**
The center of the sphere is the midpoint of the diameter. Use the midpoint formula:
\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]
2. **Calculate the Midpoint:**
Substitute the given points \((6, 7, 8)\) and \((8, -3, 19)\):
\[
\left( \frac{6 + 8}{2}, \frac{7 + (-3)}{2}, \frac{8 + 19}{2} \right) = (7, 2, 13.5)
\]
3. **Determine the Radius:**
The radius is half the distance between the two endpoints. Use the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
4. **Calculate the Distance:**
\[
\sqrt{(8 - 6)^2 + (-3 - 7)^2 + (19 - 8)^2} = \sqrt{2^2 + (-10)^2 + 11^2} = \sqrt{4 + 100 + 121} = \sqrt{225} = 15
\]
The radius \(r\) is half of 15, so \(r = 7.5\).
5. **Equation of the Sphere:**
Use the standard equation of a sphere:
\[
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
\]
Substitute \(h = 7\), \(k = 2\), \(l = 13.5\), and \(r = 7.5\):
\
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning