A mixing tank initially contains pure water with no dissolved salt

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

200\text{-L}

mixing tank initially contains pure water with no dissolved salt. Brine containing

2\text{ g/L}

of salt flows into the tank at a constant rate of

5\text{ L/min}

, and the well–stirred mixture is pumped out at the same rate so that the volume in the tank remains

200\text{ L}

. Let

y(t)

be the mass of salt (in grams) in the tank at time

t

minutes. (1) Show that the rate of change of

y(t)

can be written in the form

\dfrac{dy}{dt}=10-\dfrac{y}{40}

. Clearly state the meaning of each term in your model. (2) Solve the differential equation for

y(t)

, given that

y(0)=0

. (3) Determine the equilibrium mass of salt in the tank and verify that your solution approaches this value as

t\to \infty

. Interpret this equilibrium in terms of concentration in the tank. (4) Estimate the time required for the concentration of salt in the tank to be within

1\%

of its equilibrium concentration. Show all work and justify any inequalities you use. (5) Compute the instantaneous rate (in g/min) at which salt is leaving the tank exactly

10

minutes after the process begins, and interpret this value physically.

Related theory

Browse the problem library