(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 2 gal per minute. Pure water added instead of brine. 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 Pure water is added into the tank, therefore the salt entered is 0lb/gal The rate of salt entered can be calculated as follows: salt in = ( 2gal/min ) ( 0lb/gal ) = 0 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 = − A V × 2 = − 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 = 2 − 2 = 0 Integrate both sides. V = c 1 (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 21E

(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 20E

Chapter 10.4, Problem 22E

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