(a) To determine The expression for the amount of salt in the tank at time t . Explanation Concept used: The general solution of linear differential equation d y d x + P y = Q can be calculated as follows: y ( I . F ) = ∫ Q ( I . F ) d x Here, I . F is the integrating factor of the linear equation and it can be calculated as follows: I . F = e ∫ P d x Given: The capacity of the tank is 100 gal of water that contains 20 lb of dissolve salt. The flowing rate of the brine solution is 3 gal per minute. The brine solution entering the tank has a salt concentration of 2 lb per gal. Calculation: Consider A as the amount of salt present in the tank at time t . The rate of change of A is a net result of salt added due to continuous inlet of brine solution and salt lost due to continuous outlet of solution. It can be expressed as follows: rate of change = salt in − salt out The brine solution is entering the tank has a salt concentration of 2 lb per gallon while the outlet concentration is time varying. The rate of salt entered can be calculated as follows: salt in = ( 3gal/min ) ( 2lb/gal ) The rate of salt out can be calculated as follows: salt out = ( 2 gal/min ) ( A V lb/gal ) The differential equation of the solution can be written as follows: d A d t = 3 × 2 − A V × 2 = 6 − 2 A V The change in the volume d v d t is the net result of rate of liquid in and rate of liquid out. d v d t = 3 − 2 = 1 Integrate both sides. V = t + c 1 ....... ( 1 ) It is given that the value of V = 100 when t = 0 . Substitute 0 for t and 100 for V in equation (1) to obtain the value of c 1 . V = t + 100 Therefore, the volume of the tank is t + 100 . The above differential equation can be written in the standard linear form. d A d t = 6 − 2 A t + 100 d A d t + 2 A t + 100 = 6 The integrating factor of the resultant linear form can be calculated as follows: IF = e ∫ 2 d t ( t + 100 ) = e 2 ln ( t + 100 ) = ( t + 100 ) 2 Multiply I (b) To determine The amount of salt after one hour. (c) To determine The salt concentration when time increases.

Calculus with Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Lial, Greenwell & Ritchey, The Applied Calculus & Finite Math Series)

11th Edition

ISBN: 9780133886832

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 10.4, Problem 19E

(a)

To determine

The expression for the amount of salt in the tank at time t.

(b)

To determine

The amount of salt after one hour.

(c)

To determine

The salt concentration when time increases.

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Chapter 10.4, Problem 18E

Chapter 10.4, Problem 20E

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Chapter 10 Solutions

Calculus with Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Lial, Greenwell & Ritchey, The Applied Calculus & Finite Math Series)

Ch. 10.1 - Find all solutions of the differential equation . Ch. 10.1 - Prob. 2YT Ch. 10.1 - Prob. 3YT Ch. 10.1 - Prob. 4YT Ch. 10.1 - Prob. 1WE Ch. 10.1 - Prob. 2WE Ch. 10.1 - Prob. 3WE Ch. 10.1 - Prob. 4WE Ch. 10.1 - Prob. 5WE Ch. 10.1 - Find the general solution for each differential...

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