# Math Help - Differential Equation

1. ## Differential Equation

A tank contains kg of salt and L of water. A solution of a concentration kg of salt per liter enters a tank at the rate L/min. The solution is mixed and drains from the tank at the same rate.

Find the amount of salt in the tank after hours.

I got something like 36 kg and my equation was

A= (320e^(-8t/1000)-320)/-8

but i dont think this is correct

2. Let C be the concentration of salt in the tank.
Each minute, 0.04 kg/L * 8L = 0.32 kilos of salt goes in and C kg/L * 8 L = 8C goes out.
$\frac{dC}{dt} = 0.32 - C*8$
$\frac{dC}{0.32 - C*8} = dt$
$-ln(0.32 - C*8) = t + K$
$ln(0.32 - C*8) = -t + K$
$ln(0.32 - C*8) = -t + K$
$0.32 - C*8 = e^{-t + K}$
$C = 0.04 - Ke^{-t}$
Find K with initial condition C(0) = 0.08
$C = 0.04 + 0.04e^{-t}$
After 5 hours, C = 0.4 + almost nothing so Q = C*1000 = 40kg