This is a homework problem which I have already solved, but I want to know why my first approach did not work. I have already asked my professor this question, but he did not explain it very well.
Problem
The answer for the time is 5.78 min.
Initially I approached this by deriving the following equation while considering the movement of the salt into and out of the tank:
$m(t) = m_i+(\dot{m_{in}}-\dot{m_{out}})t$
$m(t)$ = current salt in tank
$m_i$ = initial salt in tank
$\dot{m_{in}}$ = rate of salt flowing in
$\dot{m_{out}}$ = rate of salt flowing out
Workings
Now since the rate of salt flowing out depends on the concentration of salt in the tank I add:
$\dot{m_{out}} = \frac{m(t)}{v}\cdot\frac{dv_{out}}{dt}$
$m(t)$ = current salt in the tank
$v$ = volume of tank
$\frac{dv_{out}}{dt}$ = rate of water flow out of tank
After subbing terms I get:
$m(t) = m_i+((0.05)*6-\frac{m(t)}{50}*6)t$
Since the concentration of salt in the tank is just $\frac{m(t)}{50} = C$
$c(t) = \frac{0.5+0.3*t}{(1+\frac{6}{50}*t)50} = 0.03$
I simply divide the right side by 50 and solve for $t$ to find when the concentration gets to be $0.03kg/L$. But I end up with the wrong answer.
Question
what is the flaw in logic in my equation? To me it makes perfect sense and in other cases it will work. I always thought that if I wrote a math equation that made sense and that I thought to be true, everything should work out.