402 Ordinary Differential Equations
Exercise 13. Three small ponds each containing 10 , 000 gallons of pure water
are formed by a spring rain. Water conta ining .3 lbs/gal of salt enters pond A
at 10 gal/hr. Water evaporates from pond A at 3 gal/hr and flows into pond
Bat 7 gal/hr. Water evaporates from pond Bat 2 gal/hr and flows into pond
C at 5 gal/hr. Water evaporates from pond C at 2.5 gal/hr and flows out of
the system at 2.5 gal/hr. Find the amount of salt in each pond as a function
of time. What is the limiting amount of salt in each pond?
Exercise 14. Three 150-gallon tanks initially contain pure water. Brine with
a concentration of 4 lbs/gal flows into tank A at a rate of 10 gal/min. Water
is pumped from tank A into tank Bat a rate of 6 gal/min and into tank Cat
a rate of 4 gal/min. Water is pumped from tank C into tank B at a rate of
4 gal/min. And water is pumped out of the system from tank B at the rate
of 10 gal/min. Find the amount of salt in each tank as a function of time.
Pollution in the Great Lakes As we stated earlier, one of the major
problems facing industrialized nations is water pollution. Rivers and lakes
become polluted with various types of waste products such as DDT, mercury,
and phosphorus which kill plant and marine life. Once pollution in a river is
stopped, the river cleans itself fairly rapidly. However, as this example for the
Great Lakes will illustrate, large lakes require much longer to clean themselves
by the natural process of clean water flowing into the lake and polluted water
flowing out. Figure 9.11 shows the Great Lakes, their volumes, and inflow
and outflow rates.
v = 1180
m+--R=15
h
R = 14
o!r 0 = 99
Figure 9.11 Volumes (mi^3 ) and Flow Rates (mi^3 /yr) for the Great Lakes
A simple mathematical model for pollution in the Great Lakes treats this
system of lakes as a standard, perfect mixing problem. Thus, we make the
following four assumptions: