Question

A tank contains 10 liters of pure water. Saline solution with a variable concentration 5 grams of salt per liter is pumped into the tank at rate 4 liter per minute. The tank is kept perfectly mixed and also drains at a rate of 4 liter per minute, so the volume stays constant. State the initial value problem modeling Q(t), the amount of grams of salt dissolved in the solution in the tank at time t minutes.

Answer 1

dQ(t)/dt = 20 - 2Q(t)/5 , Q(0) = 0

Explanation:

The mass flow rate dQ(t)/dt = mass flowing in - mass flowing out

Since 5 g/L of salt is pumped in at a rate of 4 L/min, the mass flow in is thus 5 g/L × 4 L/min = 20 g/min.

Let Q(t) be the mass present at any time, t. The concentration at any time ,t is thus Q(t)/volume = Q(t)/10. Since water drains at a rate of 4 L/min, the mass flow out is thus, Q(t)/10 g/L × 4 L/min = 2Q(t)/5 g/min.

So, dQ(t)/dt = mass flowing in - mass flowing out

dQ(t)/dt = 20 g/min - 2Q(t)/5 g/min

Since the salt just begins to be pumped in, the initial mass of salt in the tank is zero. So Q(0) = 0

So, the initial value problem is thus

dQ(t)/dt = 20 - 2Q(t)/5 , Q(0) = 0