A brine solution of salt flows at a constant rate of 4 ​L/min into a large tank that initially held 100 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 3 ​L/min. If the concentration of salt in the brine entering the tank is 0.2 ​kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.1 ​kg/L?

### Part 1: Determining the mass of salt in the tank after \( t \) minutes

Let \( x(t) \) represent the mass of salt in the tank (in kg) at time \( t \). The inflow of brine introduces salt into the tank, while the outflow removes the mixture of water and salt.

#### Information given:
- Inflow rate: 4 L/min
- Outflow rate: 3 L/min
- Initial volume of water: 100 L
- Concentration of salt in the inflowing brine: 0.2 kg/L
- Initial salt in the tank: 0 kg (pure water initially)

#### Step 1: Volume of solution in the tank at time \( t \)
Since the inflow rate is 4 L/min and the outflow rate is 3 L/min, the net rate of volume change is:
\[
\text{Net volume change} = (4 - 3) = 1 \, \text{L/min}.
\]
So, the total volume \( V(t) \) in the tank after \( t \) minutes is:
\[
V(t) = 100 + t \, \text{L}.
\]

#### Step 2: Rate of change of salt in

Explore how to solve a brine tank problem involving differential equations to determine the mass of salt in the tank over time. Learn the step-by-step process of using fluid dynamics, rate of change, and exponential decay concepts to calculate when the salt concentration reaches 0.1 kg/L. Suitable for high school students in Grades 11-12.

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