Thank you A tank contains 350 liters of fluid in which 30 gr

Thank you

A tank contains 350 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 5 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.

Solution

Fluid is being pumped in and pumped out at same rate
So there is always 350L of fluid in tank
At time t, the amount of salt in tank is A(t) (in grams)
So concentrations of salt at time t is A/350 g/L

Amount of brine pumped in: 5L (with salt concentration = 1 g/L)
Amount of salt pumped in: 5L * 1g/L = 5g

Amount of brine pumped out at time t: 5L (with salt concentration = A/350 g/L)
Amount of salt pumped out at time t: 5L * A/350 g/L = A/70 g

dA/dt = 5 - A/70
70 dA/dt = 350 - A
70/(350 - A) dA = dt
-70 ln|A-350| = t + C
ln|A-200| = -t/70 + C .... where C = -C/70
A - 200 = C e^(-t/70) ..... where C = e^C
A = 350 + C e^(-t/70)

Initially, brine contains 30 g of salt
A(0) = 30
350 + C e^0 = 30
C = -320

A(t) =350 - 320 e^(-t/70)