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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 the water leaving the tank per second to CA√2gh, where c (0 < c < 1) is an empirical constant. Determine a differential equation for the heighth of water at time t for the cubical tank in the figure below. The radius of the hole is 5 in., g = 32 ft/s². dh dt II circular hole Aw T 10 ft ft/s

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 the water leaving the tank per second to CA√2gh, where c (0 < c < 1) is an empirical constant. Determine a differential equation for the heighth of water at time t for the cubical tank in the figure below. The radius of the hole is 5 in., g = 32 ft/s². dh dt II circular hole Aw T 10 ft ft/s

Advanced Engineering Mathematics
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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**Text Transcription for Educational Website:** 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 the water leaving the tank per second to \(cA_h \sqrt{2gh}\), where \(c\) \((0 < c < 1)\) is an empirical constant. Determine a differential equation for the height \(h\) of water at time \(t\) for the cubical tank in the figure below. The radius of the hole is \(5\) in., \(g = 32\) ft/s\(^2\). - **Diagram Explanation:** - The diagram shows a cubical tank with water. - The top of the water level is labeled \(A_w\). - The height of the water from the bottom to the current level is represented as \(h\). - The tank has a circular hole at the bottom labeled "circular hole." - The height of the tank from top to bottom is shown as \(10\) ft. **Equation:** \[ \frac{dh}{dt} = \boxed{} \, \text{ft/s} \] (Note: The box in the equation is meant to be filled in with the appropriate expression when solved.)
Transcribed Image Text:**Text Transcription for Educational Website:** 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 the water leaving the tank per second to \(cA_h \sqrt{2gh}\), where \(c\) \((0 < c < 1)\) is an empirical constant. Determine a differential equation for the height \(h\) of water at time \(t\) for the cubical tank in the figure below. The radius of the hole is \(5\) in., \(g = 32\) ft/s\(^2\). - **Diagram Explanation:** - The diagram shows a cubical tank with water. - The top of the water level is labeled \(A_w\). - The height of the water from the bottom to the current level is represented as \(h\). - The tank has a circular hole at the bottom labeled "circular hole." - The height of the tank from top to bottom is shown as \(10\) ft. **Equation:** \[ \frac{dh}{dt} = \boxed{} \, \text{ft/s} \] (Note: The box in the equation is meant to be filled in with the appropriate expression when solved.)
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