$(4)$ A large tank is filled to capacity with $200$ gallons of pure water. A solution containing $2$ pounds of salt per gallon is pumped into the tank at a rate of $5ga\frac{l}{m}in.$ The well$-$mixed solution is pumped out at a rate of $4ga\frac{l}{m}in.$ Let $A(t)$ be the number of pounds of salt in the tank at time $t.$

$($a$)$ Determine the rate in and rate out equations.

$($b$)$ Write an equation for $\frac{dA}{dt}$ using your answers to part $($a$).$

$($c$)$ Find an equation for $A(t).$