A brine solution of salt flows at a constant rate of 

66 

​L/min into a large tank that initially held 

100100 

L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 

55 

​L/min. If the concentration of salt in the brine entering the tank is 

0.50.5 

​kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 

0.10.1 

​kg/L?




### Problem Setup:
We have a tank that initially contains pure water, and a brine solution with a salt concentration of \( 0.5 \, \text{kg/L} \) flows into the tank at a rate of \( 66 \, \text{L/min} \). The solution inside the tank is well-stirred, and it flows out at \( 55 \, \text{L/min} \). The problem asks to determine:

1. The mass of salt in the tank after \( t \) minutes.
2. When the concentration of salt in the tank reaches \( 0.1 \, \text{kg/L} \).

### Step 1: Define Variables

- Let \( x(t) \) represent the mass of salt (in kg) in the tank at time \( t \) (in minutes).
- The volume of the tank at any time \( t \) is increasing due to the inflow and outflow difference. The net inflow rate is \( 66 - 55 = 11 \, \text{L/min} \).

So, the volume of the solution at time \( t \) is:
\[
V(t) = 100100 + 11t \, \text{L}
\]

### Step 2: Differential Equation for the Mass of Salt

#### Salt Inflow:
- The salt enters the tank at a rate of:
\[
\text{Rate of salt in} = 66 \, \text{L/min} \times 0.5 \, \text{kg/L} = 33 \, \text{kg/min}
\]

#### Salt Outflow:
- The concentration of salt in the tank at time \( t \) is:
\[
\frac{x(t)}{V(t)} \, \text{kg/L}
\]
- So, the salt leaves the tank at a rate of:
\[
\text{Rate of salt out} = 55 \, \text{L/min} \times \frac{x(t)}{V(t)} = 55 \times \frac{x(t)}{100100 + 11t} \, \text{kg/min}
\]

### Step 3: Form the Differential

A brine solution of salt flows at a constant rate of 66 L/min into a large tank that initially held 100100 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 55 L/min. If the concentration of salt in the brine entering the tank is 0.5 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?

Learn how to solve a differential equation for a brine tank problem where salt flows into a tank at a rate of 66 L/min, initially containing pure water. The solution covers the rate of change, inflow, and outflow of salt in the tank, determining when the salt concentration reaches 0.1 kg/L. Suitable for undergraduate calculus students.

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