A very large tank initially contains 100L of pure water. Starting at time t=0 a solution with a salt concentration of 0.7kg/L is added at a rate of 6L/min. The solution is kept thoroughly mixed and is drained from the tank at a rate of 4L/min. Answer the following questions.1. Let y(t) be the amount of salt (in kilograms) in the tank after t minutes. What differential equation does y satisfy? Use the variable y for y(t).Answer (in kilograms per minute): dydt =
Question
A very large tank initially contains 100L of pure water. Starting at time t=0 a solution with a salt concentration of 0.7kg/L is added at a rate of 6L/min. The solution is kept thoroughly mixed and is drained from the tank at a rate of 4L/min. Answer the following questions.1. Let y(t) be the amount of salt (in kilograms) in the tank after t minutes. What differential equation does y satisfy? Use the variable y for y(t).Answer (in kilograms per minute): dydt =
Solution
The rate of change of the amount of salt in the tank is given by the difference between the rate at which salt is added to the tank and the rate at which salt is drained from the tank.
The rate at which salt is added to the tank is the product of the rate at which the solution is added and the concentration of salt in the solution, which is 6 L/min * 0.7 kg/L = 4.2 kg/min.
The rate at which salt is drained from the tank is the product of the rate at which the solution is drained and the concentration of salt in the solution in the tank at time t. The concentration of salt in the solution in the tank at time t is y(t) / (100 L + (6 L/min - 4 L/min) * t), since the volume of the solution in the tank at time t is the initial volume plus the net rate at which solution is added times time. The rate at which salt is drained is then 4 L/min * y(t) / (100 L + 2t L/min).
Therefore, the differential equation that y satisfies is:
dy/dt = 4.2 kg/min - 4 L/min * y(t) / (100 L + 2t L/min)
Simplifying the units gives:
dy/dt = 4.2 - (4/100 + 2t) * y
So, the differential equation is dy/dt = 4.2 - (0.04t + 0.02) * y.
Similar Questions
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