Answer:
The differential equation for the amount of salt A(t) in the tank at a time t > 0 is
.
Step-by-step explanation:
We are given that a large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min.
The concentration of the solution entering is 4 lb/gal.
Firstly, as we know that the rate of change in the amount of salt with respect to time is given by;
where,
= concentration of salt in the inflow
input rate of brine solution
and
= concentration of salt in the outflow
outflow rate of brine solution
So,
= 4 lb/gal
3 gal/min = 12 lb/gal
Now, the rate of accumulation = Rate of input of solution - Rate of output of solution
= 3 gal/min - 2 gal/min
= 1 gal/min.
It is stated that a large mixing tank initially holds 500 gallons of water, so after t minutes it will hold (500 + t) gallons in the tank.
So,
= concentration of salt in the outflow
outflow rate of brine solution
=
= ![\frac{2A(t)}{500+t} \text{ lb/min }](https://tex.z-dn.net/?f=%5Cfrac%7B2A%28t%29%7D%7B500%2Bt%7D%20%5Ctext%7B%20lb%2Fmin%20%7D)
Now, the differential equation for the amount of salt A(t) in the tank at a time t > 0 is given by;
=
or
.