Differential Equations $(7)$ Compartmental models: salt$-$tank dynamics.

A $400-$gal tank initially contains $100$ gal of brine containing $50$ lb of salt. Brine containing $1$ lb of salt per gallon enters the tank at the rate of $5ga\frac{l}{s},$ and the well$-$mixed brine in the tank flows out at a rate of $3ga\frac{l}{s}.$

a$)$ Find the volume $V(t)$ of liquid in the tank as a function of time.

b$)$ Formulate an appropriate ODE for $x(t),$ the amount of salt in the tank at time $t,$ with an initial condition.

c$)$ Solve the ODE using the integrating factor approach.

d$)$ How much salt will the tank contain when it is full of brine?