Tank-Modeling Problem: Solving for Salt Amount with Differential Equations

[jdm4897]

Homework Statement

A large tank initially contains 30 gallons of water and 5 pounds of salt. A salt solution consisting of 1 lb/gal is pumped into the tank at a rate of 3 gal/min, and the well-mixed solution is pumped out at the same rate. At time t = 10 an large hopper begins dumping salt into the tank at a rate of 5 lbs/min. At time t = 20, the hopper starts making an alarming noise, and the flow of salt stops. Then at time t = 30, the hopper ruptures and dumps 10 lbs of salt into the tank all at once. Find the function which gives the amount of salt in the tank at time t

Homework Equations

The differential equation for the amount of salt in tank= rate in - rate out

The Attempt at a Solution

I tried to use a heaviside function as well as the dirac delta at the end of the equation. I got stuck trying to set it up with heaviside step functions

 

[vela]

Show your work.