Explanation Given: The tank in figure 7(B) is a cylinder of radius 4 m and height 15 m . Assume that the tank is half-filled with water and that water leaks through a hole in the bottom of area B = 0.001 m 2 . Calculation: Length of the upper surface of water is L = 15 m . So, width of the upper surface of water is: W = 2 R 2 − ( R − y ) 2 = 2 R 2 − ( R 2 + y 2 − 2 R y ) = 2 R 2 − R 2 − y 2 + 2 R y = 2 2 R y − y 2 So, the area of cross-section of the upper surface of water at a distance y from the bottom is: A ( y ) = L × W = 30 2 R y − y 2 According to Toricelli’s Law, the velocity of the water flowing through a hole is: v = − 2 g y , where g = 9.8 m s 2 . So, the differential equation for the height of the water in the tank is: d y d t = B v ( y ) A ( y ) , where A ( y ) is the cross-sectional area of the tank at height y . Now, d y d t = B v ( y ) A ( y ) = B ( − 2 g y ) 30 2 R y − y 2 = B ( − 2 g ) 30 2 R − y = − 0.001 2 × 9.8 30 8 − y = − 0.0001 19.6 3 8 − y So, the separation of variables and then integrating: 8 − y d y = − 0...

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 9.1, Problem 47E

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We need to determine the water level $y (t)$ and the time $t_{e}$ when the tank is empty.

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Chapter 9.1, Problem 46E

Chapter 9.1, Problem 48E

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Chapter 9 Solutions

Calculus: Early Transcendentals

Ch. 9.1 - Prob. 1PQ Ch. 9.1 - Prob. 2PQ Ch. 9.1 - Prob. 3PQ Ch. 9.1 - Prob. 4PQ Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E

Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E Ch. 9.1 - Prob. 11E Ch. 9.1 - Prob. 12E Ch. 9.1 - Prob. 13E Ch. 9.1 - Prob. 14E Ch. 9.1 - Prob. 15E Ch. 9.1 - Prob. 16E Ch. 9.1 - Prob. 17E Ch. 9.1 - Prob. 18E Ch. 9.1 - Prob. 19E Ch. 9.1 - Prob. 20E Ch. 9.1 - Prob. 21E Ch. 9.1 - Prob. 22E Ch. 9.1 - Prob. 23E Ch. 9.1 - Prob. 24E Ch. 9.1 - Prob. 25E Ch. 9.1 - Prob. 26E Ch. 9.1 - Prob. 27E Ch. 9.1 - Prob. 28E Ch. 9.1 - Prob. 29E Ch. 9.1 - Prob. 30E Ch. 9.1 - Prob. 31E Ch. 9.1 - Prob. 32E Ch. 9.1 - Prob. 33E Ch. 9.1 - Prob. 34E Ch. 9.1 - Prob. 35E Ch. 9.1 - Prob. 36E Ch. 9.1 - Prob. 37E Ch. 9.1 - Prob. 38E Ch. 9.1 - Prob. 39E Ch. 9.1 - Prob. 40E Ch. 9.1 - Prob. 41E Ch. 9.1 - Prob. 42E Ch. 9.1 - Prob. 43E Ch. 9.1 - Prob. 44E Ch. 9.1 - Prob. 45E Ch. 9.1 - Prob. 46E Ch. 9.1 - Prob. 47E Ch. 9.1 - Prob. 48E Ch. 9.1 - Prob. 49E Ch. 9.1 - Prob. 50E Ch. 9.1 - Prob. 51E Ch. 9.1 - Prob. 52E Ch. 9.1 - Prob. 53E Ch. 9.1 - Prob. 54E Ch. 9.1 - Prob. 55E Ch. 9.1 - Prob. 56E Ch. 9.1 - 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Knowledge Booster

We need to determine the water level y ( t ) and the time t e when the tank is empty.

We need to determine the water level y(t) and the time te when the tank is empty.

Want to see the full answer?

Chapter 9 Solutions

We need to determine the water level $y (t)$ and the time $t_{e}$ when the tank is empty.