A vertical right circular cylindrical tank measures 21 feet high and has a 10 foot diameter. It is full of oil weighing 61.7 lb ft3 a) Set up the work integral [clean up to just before integrating] showing how much work (in ft-lb) it takes to pump the oil to the level of the top of the tank (to empty the tank). b) Evaluate the answer from part a. Round your answer to the nearest whole ft-lb.
A vertical right circular cylindrical tank measures 21 feet high and has a 10 foot diameter. It is full of oil weighing 61.7 lb ft3 a) Set up the work integral [clean up to just before integrating] showing how much work (in ft-lb) it takes to pump the oil to the level of the top of the tank (to empty the tank). b) Evaluate the answer from part a. Round your answer to the nearest whole ft-lb.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
![### Problem Statement
A vertical right circular cylindrical tank measures 21 feet high and has a 10-foot diameter. It is full of oil weighing 61.7 lb/ft³.
a) Set up the work integral [clean up to just before integrating] showing how much work (in ft-lb) it takes to pump the oil to the level of the top of the tank (to empty the tank).
b) Evaluate the answer from part a. Round your answer to the nearest whole ft-lb.
### Solution
To determine the work needed to pump the oil to the top of the tank, we can use the principles of calculus, specifically integration.
#### a) Setting up the Work Integral
1. **Determine the volume of a thin horizontal slice of oil**:
- Thickness \(dy\)
- Radius \(r = 5 \) feet (since the diameter is 10 feet)
- Volume \(dV = \pi r^2 dy = \pi (5)^2 dy = 25\pi dy\)
2. **Determine the weight of the thin slice**:
- Weight density of oil: \(61.7 \frac{lb}{ft^3}\)
- Weight \( = 61.7 \cdot 25\pi dy = 1542.5 \pi dy\)
3. **Determine the distance the slice must be lifted**:
- If \(y\) is the height above the bottom of the tank, then the slice must be lifted a distance of \(21 - y\) feet.
4. **Work for the slice**:
- \(dW = \text{weight} \cdot \text{distance} = 1542.5 \pi (21 - y) dy\)
5. **Set up the integral for total work**:
- Integrate from \(y = 0\) to \(y = 21\):
\[
W = \int_{0}^{21} 1542.5 \pi (21 - y) dy
\]
#### b) Evaluating the Integral
The integral is:
\[
W = 1542.5 \pi \int_{0}^{21} (21 - y) dy
\]
Integrate:
\[
W = 1542.5 \pi \left[ 21y - \frac{y^2}{2} \right]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a617c73-16c8-405c-b9c4-72dd4741d892%2Ff4642bee-177b-48d5-9b22-36680bc9091c%2Fp4jjve_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
A vertical right circular cylindrical tank measures 21 feet high and has a 10-foot diameter. It is full of oil weighing 61.7 lb/ft³.
a) Set up the work integral [clean up to just before integrating] showing how much work (in ft-lb) it takes to pump the oil to the level of the top of the tank (to empty the tank).
b) Evaluate the answer from part a. Round your answer to the nearest whole ft-lb.
### Solution
To determine the work needed to pump the oil to the top of the tank, we can use the principles of calculus, specifically integration.
#### a) Setting up the Work Integral
1. **Determine the volume of a thin horizontal slice of oil**:
- Thickness \(dy\)
- Radius \(r = 5 \) feet (since the diameter is 10 feet)
- Volume \(dV = \pi r^2 dy = \pi (5)^2 dy = 25\pi dy\)
2. **Determine the weight of the thin slice**:
- Weight density of oil: \(61.7 \frac{lb}{ft^3}\)
- Weight \( = 61.7 \cdot 25\pi dy = 1542.5 \pi dy\)
3. **Determine the distance the slice must be lifted**:
- If \(y\) is the height above the bottom of the tank, then the slice must be lifted a distance of \(21 - y\) feet.
4. **Work for the slice**:
- \(dW = \text{weight} \cdot \text{distance} = 1542.5 \pi (21 - y) dy\)
5. **Set up the integral for total work**:
- Integrate from \(y = 0\) to \(y = 21\):
\[
W = \int_{0}^{21} 1542.5 \pi (21 - y) dy
\]
#### b) Evaluating the Integral
The integral is:
\[
W = 1542.5 \pi \int_{0}^{21} (21 - y) dy
\]
Integrate:
\[
W = 1542.5 \pi \left[ 21y - \frac{y^2}{2} \right]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
