solution_2.3 - Solutions to selected problems in Section...

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Unformatted text preview: Solutions to selected problems in Section 2.3 1. Let Q ( t ) be the amount of salt in the tank measured in lb, and let t be measured in minutes. We have the following information to set up the the IVP: c i = 0 . 2 , r i = 3 , r o = r i = 3 , (flows out at the same rate) , Q (0) = 0 , fresh water , V (0) = 100 . Since r i = r o , we have V ( t ) = V (0) = 100 , and the salt concentration in the tank is c o = Q ( t ) V ( t ) = Q ( t ) 100 . Then using the conservation of salt law to set up the ODE for Q ( t ): Q ( t ) = r i c i- r o c o = 0 . 6- Q ( t ) 100 3 . Hence the IVP for Q ( t ) is Q ( t ) = 0 . 6- Q ( t ) 100 3 , Q (0) = 0 . Solving this IVP to have Q ( t ) = 20 + C exp(- 3 100 t ) . Applying the initial condition, we have 20 + C exp(- 3 100 0) = Q (0) = 0 C =- 20 and we obtain the formula for the amount of salt in the tank at any time t : Q ( t ) = 20- 20 exp(- 3 100 t ) . (a). The salt in the tank after 10 min is Q (10) = 20- 20 exp(- 3 10 ) = 5 . 184 . 1 (b). As t , Q ( t ) 20. The limiting concentration is 20 lb/100 gal or 0.2 lb/gal, same as the concentration at inflow. 4. Let Q ( t ) be the amount of salt in the tank measured in lb, and let t be measured in minutes. We have the following information to set up the the IVP: Q (0) = 5 , V (0) = 200 , c i = 0 . 1 , r i = r, (the constant rate r to be determined) r o = r i = r, (flows out at the same rate) ....
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solution_2.3 - Solutions to selected problems in Section...

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