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
![**Problem Statement:**
A ball of radius 12 has a round hole of radius 5 drilled through its center. Find the volume of the resulting solid.
**Solution:**
To solve this problem, we need to calculate the volume of the original sphere and subtract the volume of the cylindrical hole drilled through it.
1. **Volume of the Sphere:**
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
where \( r = 12 \).
2. **Volume of the Cylinder (the hole):**
\[
V_{\text{cylinder}} = \pi (r_{\text{hole}})^2 h
\]
- \( r_{\text{hole}} = 5 \)
- The height (\( h \)) of the cylinder can be found using the Pythagorean theorem in relation to the radius of the sphere: \( h = 2 \sqrt{r^2 - (r_{\text{hole}})^2} \).
Plug in the values to find \( h \) and then calculate the cylindrical volume.
3. **Volume of the Resulting Solid:**
\[
V_{\text{solid}} = V_{\text{sphere}} - V_{\text{cylinder}}
\]
**Instructional Video:**
For additional help, a video explanation is available to walk you through each calculation step visually.
**Submission:**
Once you have calculated the volume based on the above formulas, enter your answer in the provided space and click "Submit Question".
---
This section provides a clear statement of the problem along with a structured approach to finding the solution, enhancing the educational value of the content.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ce7f870-07de-467c-9b3f-39755df61668%2Ffa189d62-7bc8-4adb-bdc0-d66810239331%2Fwpoxwfi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A ball of radius 12 has a round hole of radius 5 drilled through its center. Find the volume of the resulting solid.
**Solution:**
To solve this problem, we need to calculate the volume of the original sphere and subtract the volume of the cylindrical hole drilled through it.
1. **Volume of the Sphere:**
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
where \( r = 12 \).
2. **Volume of the Cylinder (the hole):**
\[
V_{\text{cylinder}} = \pi (r_{\text{hole}})^2 h
\]
- \( r_{\text{hole}} = 5 \)
- The height (\( h \)) of the cylinder can be found using the Pythagorean theorem in relation to the radius of the sphere: \( h = 2 \sqrt{r^2 - (r_{\text{hole}})^2} \).
Plug in the values to find \( h \) and then calculate the cylindrical volume.
3. **Volume of the Resulting Solid:**
\[
V_{\text{solid}} = V_{\text{sphere}} - V_{\text{cylinder}}
\]
**Instructional Video:**
For additional help, a video explanation is available to walk you through each calculation step visually.
**Submission:**
Once you have calculated the volume based on the above formulas, enter your answer in the provided space and click "Submit Question".
---
This section provides a clear statement of the problem along with a structured approach to finding the solution, enhancing the educational value of the content.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.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