Give an adequate mathematical model for this scenario. Explain what each of your variables means. (ii) What is the concentration of the incoming salt solution?

Question

A tank contains 10 liters of pure water. A solution of an unknown, but constant concentration of salt is flowing in at 0.5 liters per minute. The water is mixed well and drained at 1 liter per minute. After 20 minutes there are 10 grams of salt in the tank. (i) Give an adequate mathematical model for this scenario. Explain what each of your variables means. (ii) What is the concentration of the incoming salt solution?

Answer 1

Let q(t)grams is mass of salt in tank in moment t.

Then change of mass of salt dq/dt = 0.5c - 1q/(10 - 0.5t);

q' + (1/(10 - 0.5t))q = 0.5c, c g/L is unknown concentration.

We get linear ODE. Integrated factor M = e^{∫1/(10 - 0.5t)dt} = e^{- 2ln(10 - 5t)} = (10 - 0.5t)^-2.

Multiplying by m our ODE: (10 - 0.5t)^-2q' + (10 - 0.5t)^-3q = 0.5c(10 - 0.5t)^-2;

[(10 - 0.5t)^-2q]' = 0.5c(10 - 0.5t)^-2; (10 - 0.5t)^-2q = ∫0.5c(10 - 0.5t)^-2dt;

(10 - 0.5t)^-2q = c(10 - 0.5t)^-1 + N; q = c(10 - 0.5t) + N(10 - 0.5t)²; at t = 0 q= 0; 0 = 10c + 100N, so N = - 0.1c

At last q = c(10 - 0.5t) - 0.1c(10 - 0.5t)²; If t = 20 min q = 10 g;

But by this data c is undefined.