A large open $500-$gal tank is partially filled with $100$ gallons of fluid in which $10$ pounds of salt is dissolved. Brine containing $\frac{1}{2}$ pound of salt per gallon is pumped into the tank at a rate of $4ga\frac{l}{m}in.$ The well$-$mixed solution is then pumped out at a slower rate of $2ga\frac{l}{m}in.$ Derive the engineering design equation that will enable you to calculate the number of pounds of salt in the tank at any time $t$ and calculate the concentration of salt in the tank after $30$ minates. Over what time period will your design equation be valid and why?