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Homework $2\mathrm{.}1$ The Method of the...

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A tank contains $50$ kg of salt and $1000$ L of water. A solution of a concentration $0\mathrm{.}025$ kg of salt per liter enters a tank at the rate $8\frac{L}{m}in.$ The solution is mixed and drains from the tank at the same rate.

$($a$)$ What is the concentration of our solution in the tank initially?

concentration $=$

$($kg$/$L$)$

$($b$)$ Set up an initial value problem for the quantity $y,$ in kg $,$ of salt in the tank at time t minutes.

$\frac{dy}{dt}=\mathrm{.}2-\frac{y}{125}(k\frac{g}{m}in)$

$y(0)=50(kg)$

$($c$)$ Solve the initial value problem in part $($b$).$

$y(t)=2525e^{-\frac{t}{125}}$

$($d$)$ Find the amount of salt in the tank after $4$ hours.

amount $=$

$(kg)$

$($e$)$ Find the concentration of salt in the solution in the tank as time approaches infinity.

concentration $=$

$($kg$/$L$)$