A $1000m^3$ capacity rectangular reservoir is $80\%$ filled with pure water. Staring at $7$:$00$ am

on Feb $1,$ a $5\%$ chloric acid solution flows in at $10\frac{L}{sec}$ and well$-$stirred mixture flows out

at $20\frac{L}{sec}.$

$($a$)$ Develop a model for the volume $V$ of the solution in the reservoir and the amount $A$ of

dissolved chloric acid as functions of time. Your model should specify a consistent system

of units, describe the variables and include differential equations and initial conditions.

Note: $1m^3=1000L$

$($b$)$ Solve the initial value problems in part $($a$).$

$($c$)$ At what time of the day will the volume of the liquid in the reservoir drop to $20\%$ of

reservoir's capacity? Round off the answer to the nearest minute if necessary.

$($d$)$ Determine the formula for the concentration of the chloric acid in the reservoir as a

function of time and use it to calculate the concentration of the chloric acid in the

reservoir at $5$:$00$ pm on the same day.