Suppose water is leaking from a tank through a circular hole of area A, at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to CA,V 2gh, where c (0
Suppose water is leaking from a tank through a circular hole of area A, at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to CA,V 2gh, where c (0
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%
![**Educational Content Transcription**
---
**Problem Description:**
(a) If the tank is initially full, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)
**Solution:**
14.31 ✔️ minutes
---
(b) Suppose the tank has a vertex angle of 60° and the circular hole has a radius of 4 inches. Determine the differential equation governing the height \( h \) of water. Use \( c = 0.6 \) and \( g = 32 \, \text{ft/s}^2 \).
\[
\frac{dh}{dt} = \, \text{[Box for input]}
\]
**Additional Problem:**
Solve the initial value problem that assumes the height of the water is initially 10 feet.
\[
h(t) = \, \text{[Box for input]} \, \times
\]
If the height of the water is initially 10 feet, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)
\[
\text{[Box for input]} \, \times \, \text{minutes}
\]
---
**Explanation/Instructions:**
- Each question is supplemented with a box for input, prompting learners to insert their solutions.
- Checkmarks (✔️) indicate correct input or solution.
- Crosses (✖️) indicate incorrect input or solution, suggesting the need for a revised answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F709380c1-0dd1-480a-83cd-39f2ee90eebf%2F97b2f08a-2add-467d-8510-c01f0e3e42ac%2Fniy7687_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Content Transcription**
---
**Problem Description:**
(a) If the tank is initially full, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)
**Solution:**
14.31 ✔️ minutes
---
(b) Suppose the tank has a vertex angle of 60° and the circular hole has a radius of 4 inches. Determine the differential equation governing the height \( h \) of water. Use \( c = 0.6 \) and \( g = 32 \, \text{ft/s}^2 \).
\[
\frac{dh}{dt} = \, \text{[Box for input]}
\]
**Additional Problem:**
Solve the initial value problem that assumes the height of the water is initially 10 feet.
\[
h(t) = \, \text{[Box for input]} \, \times
\]
If the height of the water is initially 10 feet, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)
\[
\text{[Box for input]} \, \times \, \text{minutes}
\]
---
**Explanation/Instructions:**
- Each question is supplemented with a box for input, prompting learners to insert their solutions.
- Checkmarks (✔️) indicate correct input or solution.
- Crosses (✖️) indicate incorrect input or solution, suggesting the need for a revised answer.
![Suppose water is leaking from a tank through a circular hole of area \( A_h \) at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to \( cA_h \sqrt{2gh} \), where \( c \) (0 < c < 1) is an empirical constant.
A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (Assume the removed apex of the cone is of negligible height and volume.)
**(a)** Suppose the tank is 20 feet high and has a radius of 8 feet, and the circular hole has a radius of 2 inches. The differential equation governing the height \( h \) in feet of water leaking from a tank after \( t \) seconds is
\[
\frac{dh}{dt} = -\frac{5}{6h^{3/2}}.
\]
In this model, friction and contraction of the water at the hole are taken into account with \( c = 0.6 \), and \( g \) is taken to be 32 ft/s².
**Diagram Description:**
The diagram shows a right-circular cone-shaped tank standing upright, with water leaking from a small circular hole at the bottom.
- The tank height is labeled as 20 ft.
- The radius of the top of the tank is labeled as 8 ft.
- There is a notation for the area of the water surface, \( A_w \).
- The height of water at any time is labeled as \( h \).
- The small circular hole at the bottom of the tank is shown with a stream of water leaking out.
The task is to solve the initial value problem assuming the tank is initially full.
The solution to the problem is:
\[
h(t) = \left(800 - \frac{25t}{12}\right)^{2/5}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F709380c1-0dd1-480a-83cd-39f2ee90eebf%2F97b2f08a-2add-467d-8510-c01f0e3e42ac%2Fyfhjqb8_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose water is leaking from a tank through a circular hole of area \( A_h \) at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to \( cA_h \sqrt{2gh} \), where \( c \) (0 < c < 1) is an empirical constant.
A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (Assume the removed apex of the cone is of negligible height and volume.)
**(a)** Suppose the tank is 20 feet high and has a radius of 8 feet, and the circular hole has a radius of 2 inches. The differential equation governing the height \( h \) in feet of water leaking from a tank after \( t \) seconds is
\[
\frac{dh}{dt} = -\frac{5}{6h^{3/2}}.
\]
In this model, friction and contraction of the water at the hole are taken into account with \( c = 0.6 \), and \( g \) is taken to be 32 ft/s².
**Diagram Description:**
The diagram shows a right-circular cone-shaped tank standing upright, with water leaking from a small circular hole at the bottom.
- The tank height is labeled as 20 ft.
- The radius of the top of the tank is labeled as 8 ft.
- There is a notation for the area of the water surface, \( A_w \).
- The height of water at any time is labeled as \( h \).
- The small circular hole at the bottom of the tank is shown with a stream of water leaking out.
The task is to solve the initial value problem assuming the tank is initially full.
The solution to the problem is:
\[
h(t) = \left(800 - \frac{25t}{12}\right)^{2/5}
\]
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 2 steps with 2 images
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,
