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√2gh 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 radius 8 feet and the circular hole has radius 2 inches. The differential equation governing the heighth in feet of water leaking from a tank after t seconds is dh 5 6h3/2 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. See the figure below. h(t) = 8 ft 20 ft Solve the initial value problem that assumes the tank is initially full. ·|(-1/52¹ + + + 800√/5) (³) circular hole dh dt If the tank is initially full, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.) 14.31 ✓ minutes (b) Suppose the tank has a vertex angle of 60° and the circular hole has radius 3 inches. Determine the differential equation governing the heighth of water. Use c = 0.6 and g = 32 ft/s². 0.5 Solve the initial value problem that assumes the height of the water is initially 10 feet. h(t) = 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.) 2:34 ✔ minutes where c (0 << < 1) is an empirical constant.
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√2gh 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 radius 8 feet and the circular hole has radius 2 inches. The differential equation governing the heighth in feet of water leaking from a tank after t seconds is dh 5 6h3/2 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. See the figure below. h(t) = 8 ft 20 ft Solve the initial value problem that assumes the tank is initially full. ·|(-1/52¹ + + + 800√/5) (³) circular hole dh dt If the tank is initially full, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.) 14.31 ✓ minutes (b) Suppose the tank has a vertex angle of 60° and the circular hole has radius 3 inches. Determine the differential equation governing the heighth of water. Use c = 0.6 and g = 32 ft/s². 0.5 Solve the initial value problem that assumes the height of the water is initially 10 feet. h(t) = 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.) 2:34 ✔ minutes where c (0 << < 1) is an empirical constant.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem: Water Leakage from a Tank
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 \( \frac{c A_h \sqrt{2gh}}{A_w} \), where \( 0 < c < 1 \) 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.)
### Part (a)
Consider a tank that is 20 feet high and has a radius of 8 feet. 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 \( 32 \, ft/s^2 \).
#### Diagram: Right-Circular Cone Tank
The diagram shows a right-circular cone with the following dimensions:
- Height: 20 feet
- Base radius: 8 feet
- Circular hole at the bottom with a 2-inch radius
#### Solution and Calculation
Solve the initial value problem that assumes the tank is initially full:
\[ h(t) = \left( -\frac{25}{12^2} + 800\sqrt{5} \right) \left( \frac{3}{2} t \right)^{2/3}. \]
If the tank is initially full, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)
Result:
\[ 14.31 \text{ minutes}. \]
### Part (b)
Suppose the tank has a vertex angle of 60° and the circular hole has a radius of 3 inches. Determine the differential equation governing the height \( h \) of the water. Use \( c = 0.6 \) and \( g = 32 \, ft/s^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29c74d06-0f3b-4eb2-9c9d-dbbc1918002c%2F7896992f-d989-4f84-a6b5-a854c6a929f9%2Fsvdvgiq_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem: Water Leakage from a Tank
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 \( \frac{c A_h \sqrt{2gh}}{A_w} \), where \( 0 < c < 1 \) 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.)
### Part (a)
Consider a tank that is 20 feet high and has a radius of 8 feet. 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 \( 32 \, ft/s^2 \).
#### Diagram: Right-Circular Cone Tank
The diagram shows a right-circular cone with the following dimensions:
- Height: 20 feet
- Base radius: 8 feet
- Circular hole at the bottom with a 2-inch radius
#### Solution and Calculation
Solve the initial value problem that assumes the tank is initially full:
\[ h(t) = \left( -\frac{25}{12^2} + 800\sqrt{5} \right) \left( \frac{3}{2} t \right)^{2/3}. \]
If the tank is initially full, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)
Result:
\[ 14.31 \text{ minutes}. \]
### Part (b)
Suppose the tank has a vertex angle of 60° and the circular hole has a radius of 3 inches. Determine the differential equation governing the height \( h \) of the water. Use \( c = 0.6 \) and \( g = 32 \, ft/s^2 \).
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