1) A tank contains 100 gallons of water in which 40 pounds of salt are dissolved. A brine solution containing 5 pounds of salt per gallon is pumped into the tank at the rate of 4 gallons per minute. The mixture is stirred well and is pumped out of the tank at the same rate. Let A(t) represent the amount of salt in the tank at time t. a) Solve this initial value problem. How much salt will be present in the tank at time t? b) Explain how you can answer the question how much salt will be in the tank after a long time without solving the differential equation.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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**Problem Statement: Salt Mixture in a Tank**

1) A tank contains 100 gallons of water in which 40 pounds of salt are dissolved. A brine solution containing 5 pounds of salt per gallon is pumped into the tank at the rate of 4 gallons per minute. The mixture is stirred well and is pumped out of the tank at the same rate. Let \( A(t) \) represent the amount of salt in the tank at time \( t \).

   a) Solve this initial value problem. How much salt will be present in the tank at time \( t \)?

   b) Explain how you can answer the question of how much salt will be in the tank after a long time without solving the differential equation.

**Explanation:**

- The problem involves a tank containing a saltwater solution, where brine is both added and removed at equal rates, maintaining constant volume. The concentration and rate of the incoming brine and the dynamics of the solution are essential to setting up and solving the differential equation for \( A(t) \).

- Consider what happens as time approaches infinity to reason about the system's behavior without computation.
Transcribed Image Text:**Problem Statement: Salt Mixture in a Tank** 1) A tank contains 100 gallons of water in which 40 pounds of salt are dissolved. A brine solution containing 5 pounds of salt per gallon is pumped into the tank at the rate of 4 gallons per minute. The mixture is stirred well and is pumped out of the tank at the same rate. Let \( A(t) \) represent the amount of salt in the tank at time \( t \). a) Solve this initial value problem. How much salt will be present in the tank at time \( t \)? b) Explain how you can answer the question of how much salt will be in the tank after a long time without solving the differential equation. **Explanation:** - The problem involves a tank containing a saltwater solution, where brine is both added and removed at equal rates, maintaining constant volume. The concentration and rate of the incoming brine and the dynamics of the solution are essential to setting up and solving the differential equation for \( A(t) \). - Consider what happens as time approaches infinity to reason about the system's behavior without computation.
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