Explanation Given: The initial volume of the tank is 300 gallons . The amount of salt dissolved initially is 50 pounds , the rate of pumping the water in the tank is 3 gal / min and the rate of pumping water out from the tank is 2 gal / min . The amount of salt in the solution finally is A at time t > 0 . Procedure used: Rate of change of amount of salt with respect to time: d A d t = R i n − R o u t Here, the rate of change of amount of salt is d A d t , the inlet flow rate is R i n and the rate of output flow rate of salt is R o u t Input flow rate of the salt : R i n = ( concentration of salt in the input flow ) × ( input rate of water ) Output flow rate of the salt: R o u t = ( concentration of salt in outlfow ) × ( output rate of water ) Rate of accumulation of the solution due to slower rate of pumping out the water from the tank : Rate of accumulation = ( Rate of the input flow of solution ) − ( Rate of outflow of the solution ) Calculation: Write the expression for rate of change of amount of salt with respect to time. d A d t = R i n − R o u t (1). Here, the rate of change of amount of salt is d A d t , the inlet flow rate is R i n and the rate of output flow rate of salt is R o u t Write the expression for input flow rate of the salt. R in = ( concentration of salt in the input flow ) × ( input rate of water ) (2) Write the expression output flow rate of the salt. R o u t = ( concentration of salt in outlfow ) × ( output rate of water ) (3) Write the expression for rate of accumulation of the solution due to slower rate of pumping out the water from the tank. Rate of accumulation = ( Rate of the input flow of solution ) − ( Rate of outflow of the solution ) = ( 3 − 2 ) gal / min = 1 gal / min Since the rate of accumulation of the solution in the tank is 1 gal / min , therefore the amount of solution in the tank after t minutes is given by ( 300 + t ) gallons . Write the expression for the concentration of salt during outflow. concentration of salt during outflow = amount of salt × 1 amount of solution in tank Substitute A for amount of salt and 300 + t for amount of solution in tank at time t in the concentration of salt during outflow equation

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 1.3, Problem 10E

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Determine a differential equation for the aunt of salt A(t) in the tank at time t>0.

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Determine a differential equation for the aunt of salt A ( t ) in the tank at time t &gt; 0 .

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Determine a differential equation for the aunt of salt A(t) in the tank at time t>0.

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Determine a differential equation for the aunt of salt A ( t ) in the tank at time t > 0 .