(A) Explanation Given: The tank can hold 1,000 gallons of water. Initially the tank contains 100 gallons of unpolluted water. Additional polluted water containing 2 pounds of pollutants per gallon is pumped into the tank at the rate of 75 gallons of water per hour. The uniformly mixed water in the tank is released into the sewage system at a rate of 50 gallons per hour. The process continues for 2 hours. Formula used: Integrating factor: The integrating factor of the First-Order linear differential equation y ′ + f ( x ) y = g ( x ) is I ( x ) = e ∫ f ( x ) d x . I ( x ) = e ∫ f ( x ) d x . Calculation: Let p ( t ) be the total amount of pollutants in the tank t hours after this process begins. Since the tank initially contains 100 gallons of unpolluted water, p ( 0 ) = 0 . Since water containing 2 pounds of pollutants per gallon is entering the tank at a constant rate of 75 gallons per hour, pollutants are entering the tank at a constant rate of 2 × 75 = 150 pounds per hour . Note that 75 gallons of water enter the tank each hour and 50 gallons leave each hour. That means the amount of water in the tank increases at the rate of 25 gallons per hour. Therefore, the amount of water in the tank at time t is 100 + 25 t . The amount of pollutants in each gallon of water at any time t is obtained by dividing the total amount of pollutants by the total amount of water. That is, p ( t ) 100 + 25 t . Since water is leaving the tank at the rate of 50 gallons per hour, the rate at which the pollutants are leaving the tank is 50 p ( t ) 100 + 25 t = 2 p ( t ) 4 + t . So the rate of change of P ( t ) must satisfy the equation p ′ ( t ) = 150 − 2 p ( t ) 4 + t where p ( 0 ) = 0 . That is p ′ ( t ) + 2 p ( t ) 4 + t = 150 . Now find the integrating factor of this first order linear equation where f ( t ) = 2 4 + t and g ( t ) = 150 . I ( t ) = e ∫ 2 4 + t d t = e 2 ln ( 4 + t ) = e ln ( 4 + t ) 2 = ( 4 + t ) 2 Multiply both sides of the equation by the integrating factor (B) To determine To find: The time at which the tank will contain 700 pounds of pollutants.

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 9, Problem 46RE

(A)

To determine

To find: The total amount of pollutants in the tank after 2 hours.

(B)

To determine

To find: The time at which the tank will contain 700 pounds of pollutants.

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Chapter 9, Problem 45RE

Chapter 9, Problem 47RE

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Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

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The total amount of pollutants in the tank after 2 hours.

Solution Summary: The author calculates the total amount of pollutants in the tank after 2 hours.