Problem Overview

We have two processes happening in sequence:

1. First Process (0 to 12 minutes): A saltwater solution containing 50 g of salt per liter enters the tank at a rate of 1 liter per minute, while a well-stirred mixture of water and salt leaves at the same rate (1 liter/min).
2. Second Process (12 to 24 minutes): Fresh water enters at a rate $r$ liters per minute, and a well-stirred mixture of water and salt leaves at the same rate $r$ liters per minute.

First Process: 0 to 12 minutes

Let $x(t)$ represent the amount of salt (in grams) in the tank at time $t$ (in minutes).

• Inflow rate of salt: Since the inflowing water contains 50 g of salt per liter and enters at 1 liter per minute, the rate at which salt enters the tank is $50 \, \text{g/min}$.
• Outflow rate of salt: The well-stirred mixture leaves at a rate of 1 liter per minute, and since the tank volume is constant at 200 liters, the concentration of salt in the tank at time $t$ is $\frac{x(t)}{200} \, \text{g/liter}$. So the outflow rate of salt is $\frac{x(t)}{200} \, \text{g/min}$.

Thus, the differential equation describing the salt content in the tank during the first process is: $\frac{dx}{dt} = 50 - \frac{x(t)}{200}$ This is a linear first-order differential equation.

Solution of the differential equation:

Rewriting the equation: $\frac{dx}{dt} + \frac{1}{200}x = 50$ The integrating factor $\mu(t)$ is: $\mu(t) = e^{\int \frac{1}{200} dt} = e^{\frac{t}{200}}$ Multiplying both sides by $e^{\frac{t}{200}}$: $e^{\frac{t}{200}} \frac{dx}{dt} + \frac{1}{200} e^{\frac{t}{200}} x = 50 e^{\frac{t}{200}}$ Integrating both sides with respect to $t$: $e^{\frac{t}{200}} x = \int 50 e^{\frac{t}{200}} dt = 50 \cdot 200 e^{\frac{t}{200}} + C$ Simplifying: $x(t) = 50 \cdot 200 + C e^{-\frac{t}{200}} = 10000 + C e^{-\frac{t}{200}}$ At $t = 0$, the tank contains only fresh water, so $x(0) = 0$: $0 = 10000 + C \implies C = -10000$ Thus, the solution for $x(t)$ during the first process is: $x(t) = 10000 \left( 1 - e^{-\frac{t}{200}} \right)$ At $t = 12$ minutes, the amount of salt in the tank is: [ x(12) = 10000 \left(