Chapter 5.2, Problem 26P

In each of Problems $23$ through $27$, find the Laplace transformation $Y\left(s\right)=L\left\{y\right\}$ of the solution of the given initial value problem. A method of determining the inverse transform is developed in Section $5.5$.

A tank originally contains $640$ gallons (gal) of fresh water. Then water containing $\frac{1}{2}$ pound (lb) of salt per gallon is poured into the tank at a rate of $8\text{gal/min}$, and the mixture is allowed to leave at the same rate. After $15\text{min}$ the salt water solution flowing into the tank suddenly switches to fresh water flowing in at a rate of $8\text{gal/min}$, while the solution continues to leave the tank at the same rate. Find the Laplace transform of the amount of salt $y\left(t\right)$ in the tank.