Rate of change and integrating factor

[nysnacc]

Homework Statement

Initially 5 grams of salt are dissolved in 20 liters of water. Brine with concentration of salt 2 grams of salt per lter is added at a rate of 3 liters per minute. The tank is mixed well and is drained at 3 liters a minute. How long does the process have to continue until there are 20 grams of salt in the tank?

Homework Equations

Rate in / Rate out

Concentration = mass / Volume

The Attempt at a Solution

Rate in = 3L /min
Concentration in =2 g/L

rate out =3L /min
Concentration out =x g/L

I set change of concentration as x'
x' = lim Δt→0 (3 L/ min * 2 g/L *Δt - 3 L/ min * x g/L *Δt) / V *Δt
so x' = (3*2 - 3*x) / V where V =20 L (given)

x' = 3/10 - 3x/20

then use integrating factor, but I have different answer what am I doing wrong? thank

 

[Ray Vickson]

Homework Statement

Initially 5 grams of salt are dissolved in 20 liters of water. Brine with concentration of salt 2 grams of salt per lter is added at a rate of 3 liters per minute. The tank is mixed well and is drained at 3 liters a minute. How long does the process have to continue until there are 20 grams of salt in the tank?

Homework Equations

Rate in / Rate out

Concentration = mass / Volume

The Attempt at a Solution

Rate in = 3L /min
Concentration in =2 g/L

rate out =3L /min
Concentration out =x g/L

I set change of concentration as x'
x' = lim Δt→0 (3 L/ min * 2 g/L *Δt - 3 L/ min * x g/L *Δt) / V *Δt
so x' = (3*2 - 3*x) / V where V =20 L (given)

x' = 3/10 - 3x/20

then use integrating factor, but I have different answer what am I doing wrong? thank

What is your answer? Maybe it is not wrong at all, but we cannot tell without seeing it.

 

[nysnacc]

What is your answer? Maybe it is not wrong at all, but we cannot tell without seeing it.

I have x(t) = 2 - 7/4 EXP (-3/20 t)
which then gives t = 3.73 min
for 20 gram salt in the water (concentration 1 g/L)

But the answer said to be 40/3 ln(4/3) which is 3.84 min