A tank is filled with 1000 liters of pure water. Brine containing 0.04 kg of salt per liter enters the tank at 6 liters per minute. Another brine solution containing 0.06 kg of salt per liter enters the tank at 6 liters per minute. The contents of the tank are kept thoroughly mixed and the drains from the tank at 12 liters per minute.A. Determine the differential equation which describes this system. Let 𝑆(𝑡) denote the number of kg of salt in the tank after 𝑡 minutes.
Question
A tank is filled with 1000 liters of pure water. Brine containing 0.04 kg of salt per liter enters the tank at 6 liters per minute. Another brine solution containing 0.06 kg of salt per liter enters the tank at 6 liters per minute. The contents of the tank are kept thoroughly mixed and the drains from the tank at 12 liters per minute.A. Determine the differential equation which describes this system. Let 𝑆(𝑡) denote the number of kg of salt in the tank after 𝑡 minutes.
Solution
To solve this problem, we need to understand that the rate of change of salt in the tank is equal to the rate of salt coming in minus the rate of salt going out.
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Rate of salt coming in: There are two sources of salt coming in. The first brine solution is coming in at a rate of 6 liters per minute with a concentration of 0.04 kg/liter, so it's bringing in salt at a rate of 6 * 0.04 = 0.24 kg/minute. The second brine solution is coming in at the same rate but with a concentration of 0.06 kg/liter, so it's bringing in salt at a rate of 6 * 0.06 = 0.36 kg/minute. Therefore, the total rate of salt coming in is 0.24 + 0.36 = 0.6 kg/minute.
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Rate of salt going out: The solution is draining from the tank at a rate of 12 liters per minute. The concentration of salt in the tank at any given time is S(t)/1000 kg/liter (since the tank always contains 1000 liters of solution due to the inflow and outflow rates being equal). Therefore, the rate of salt going out is 12 * (S(t)/1000) = 0.012*S(t) kg/minute.
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Differential equation: Now we can set up the differential equation. The rate of change of S(t) is equal to the rate of salt coming in minus the rate of salt going out, so we have:
dS/dt = 0.6 - 0.012*S(t)
This is the differential equation that describes the system.
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