Set-up an Initial Value Problem describing the above problem and compute the initial amount of salt in the tank right before the process started.

Question

A tank of max capacity of 120 liters in volume contained 40 liters of brine solution initially. Fresh water is flown in at a rate of 7 liters per minute. The solution inside the tank is well stirred and flows out at a rate of 3 liters per minute. When the tank is full, it has 11 kilograms of salt in it.

 

Set-up an Initial Value Problem describing the above problem and compute the initial amount of salt in the tank right before the process started. Please show ALL work.

tial

Answer 1

Let m is m of solt at time t, then dm/dt = - 3m/(v₀ + 4t), At t = 0 m = m₀ and v₀ + 4t = v. We separate variables:

dm/m = - -3dt/(v₀ + 4t). So, lnm = - 3/4 ln(v₀ + 4t) + lnC, m = C(v₀ + 4t)^{- 3/4}. m₀ = Cv₀^-3/4.

Now 120 = 40 + 4t, so t = 20 min. m = C(120)^-3/4 = 11, C = 11·120^3/4 = 398.82

For mass of solt we have now m = 398.82·(v₀ + 4t)^-3/4. If t = 0 m₀ = 398.82·v₀^-3/4 = 398.82/40^3/4 = 25.87 kg .