$(50$ points$)$ Suppose that a tank contains $1000$ gallons of a solution consisting of $200$ pounds of salt dissolved in water. Pure water is pumped into the tank at a rate of $6$ gallons per minute. At the same time, the tank is drained at the same rate. Assume that the brine mixture is kept well stirred.

$($a$)$ Set up an initial value problem to model the amount of salt in the tank at time $t.$

$($b$)$ How long will it take until there is only $20$ pounds of salt left in the tank?