Question

A tank contains 400 liters of fluid in which 15 grams of salt aredissolved. Brinecontaining 3 grams of salt per liter is then pumped into the tank at the rate of 8liters per minute, and the well mixed solution is then pumpedout at the slowerrate of 2 liters per minute. Write a differential equation, with initial condition,that models the amount of saltA(t)in the tank at timet.Extra Credit (6 points)

Answer 1

x´ + 2*x = 24

Initial condition is t = 0 x = 15/400 grs/lt

Explanation:

15 grams of salt in 400 liters means 15/400 or 0,0375 grams per liter

The concentration in the tank at any moment could be written as:

Δ(x)(t) = concentration (pumped in )* rate of pumping in *δt - concentration (pumped out )* rate of pumping out*δt

Δ(x)(t) = 3*8*δt - x(s) *2*δt

Dividing by δt on both sides of the equation we get:

Δ(x)(t)/ δt = 24 - 2*x(s)

We should remember that the first member of the equation when δt⇒0 is the derivative of x with respect to time then:

D(x)/dt = 24 - 2*x

or

x´ + 2*x = 24

That is a first-order differential linear equation.

Initial condition is t = 0 x = 15/400