Consider a hemispherical tank that is completely filled with water. The water exits the tank from the spout as shown. L If R = 9 m, and L = 2 m, calculate the work in megajoules (1 MJ = 1 x 106 J) required to pump all water out of the tank. The density of water is 1000 kg/m³, and g = 9.8 m/s². (Use decimal notation. Give your answer to three decimal places.)
Consider a hemispherical tank that is completely filled with water. The water exits the tank from the spout as shown. L If R = 9 m, and L = 2 m, calculate the work in megajoules (1 MJ = 1 x 106 J) required to pump all water out of the tank. The density of water is 1000 kg/m³, and g = 9.8 m/s². (Use decimal notation. Give your answer to three decimal places.)
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
![### Hemispherical Tank Water Pumping Problem
Consider a hemispherical tank that is completely filled with water. The water exits the tank from the spout as shown in the diagram.
<img src="hemispherical_tank.png" alt="Hemispherical Tank Diagram" width="300"/>
**Diagram Explanation:**
- The tank is represented as a half-sphere (hemisphere) with a radius \( R \).
- The spout extends vertically from the top of the tank, with a length \( L \).
Given:
- Radius \( R = 9 \) meters
- Length of the spout \( L = 2 \) meters
- Density of water \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \)
- Work required calculation in megajoules (1 MJ = \( 1 \times 10^6 \) J)
**Problem Statement:**
Calculate the work in megajoules required to pump all water out of the tank. Use decimal notation and provide the answer to three decimal places.
**Solution Approach:**
To solve this problem, follow these steps:
1. Determine the volume of the hemispherical tank.
2. Calculate the total mass of water in the tank.
3. Integrate the force required to lift each differential volume element of water out of the tank to the height of the spout.
4. Convert the total work from joules to megajoules.
**Formulas Needed:**
1. Volume of a hemisphere:
\[ V = \frac{2}{3} \pi R^3 \]
2. Mass of water:
\[ m = \rho \cdot V \]
3. Work required to lift a differential volume \( dV \) to height \( h \):
\[ dW = \rho g h \, dV \]
4. Total work (W) required:
\[ W = \int_0^R \rho g h \, dV(h) \]
Where:
\[ dV = \pi (R^2 - h^2) \, dh \]
\[ h \text{ is the height from the bottom of the tank up to level } x. \]
Details of the exact integration process and calculations can be included in the educational material to guide students through the process](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffddfa066-ae14-4710-b68e-f1412b1153a3%2F3a240d48-1850-40bf-aa0f-c8849a21d433%2Fjleefhp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Hemispherical Tank Water Pumping Problem
Consider a hemispherical tank that is completely filled with water. The water exits the tank from the spout as shown in the diagram.
<img src="hemispherical_tank.png" alt="Hemispherical Tank Diagram" width="300"/>
**Diagram Explanation:**
- The tank is represented as a half-sphere (hemisphere) with a radius \( R \).
- The spout extends vertically from the top of the tank, with a length \( L \).
Given:
- Radius \( R = 9 \) meters
- Length of the spout \( L = 2 \) meters
- Density of water \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \)
- Work required calculation in megajoules (1 MJ = \( 1 \times 10^6 \) J)
**Problem Statement:**
Calculate the work in megajoules required to pump all water out of the tank. Use decimal notation and provide the answer to three decimal places.
**Solution Approach:**
To solve this problem, follow these steps:
1. Determine the volume of the hemispherical tank.
2. Calculate the total mass of water in the tank.
3. Integrate the force required to lift each differential volume element of water out of the tank to the height of the spout.
4. Convert the total work from joules to megajoules.
**Formulas Needed:**
1. Volume of a hemisphere:
\[ V = \frac{2}{3} \pi R^3 \]
2. Mass of water:
\[ m = \rho \cdot V \]
3. Work required to lift a differential volume \( dV \) to height \( h \):
\[ dW = \rho g h \, dV \]
4. Total work (W) required:
\[ W = \int_0^R \rho g h \, dV(h) \]
Where:
\[ dV = \pi (R^2 - h^2) \, dh \]
\[ h \text{ is the height from the bottom of the tank up to level } x. \]
Details of the exact integration process and calculations can be included in the educational material to guide students through the process
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 with 3 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