Dilusion...Did I make any mistakes?

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• Sep 27th 2007, 01:56 AM
circuscircus
Dilusion...Did I make any mistakes?
A tank contains 1000L of brine with a concentration of .5kb of salt per liter. In order to dilute the solution, brine with a concentration of 0.1kg of salt per liter runs into the tank at a rate of 20L/min and the resulting solution, which is stirred continuously, runs out at the same rate. How many kilograms of salt will remain after t minutes?

$A = kilograms of salt$

$A' = (concentration in rate * rate of flow in) - (concentration out rate * rate of flow out)$

$A' = (0.1 * 20) - (A/1000 * 20)$

$dA/dt = 2 - 0.02A$

$dA = (2 - 0.02A ) dt$

$\int dA/(2 - 0.02A ) = \int dt$

$ln \left|2 - 0.02A\right| + C_2 = t + C_1$

$Note: C = C_1 - C_2$

$2 - 0.02A = e^{t+C}$

$0.02A = 2 -Ke^t for K \in R$

$A(t) = 50(2-Ke^t)$

$A(0) = 0.5 = 50(2-Ke^0)$

$A(0) = 0.01 = 2-K$

$A(0) = 0.01-2 = -K$

$K = 1.99 = 199/100$

$A(t) = 50(2-\frac{199}{100}e^t)$

From skimming this, did I make any mistakes?