1550078481-Ordinary_Differential_Equations__Roberts_

152 Ordinary Differential Equations

is blown into the office at a rate of .2 ft^3 /min. Assuming the air in the

room is uniformly mixed and leaves the room at the same rate, at what

time does the amount of carbon monoxide in the air reach .0153? (Pro-

longed exposure to this concentration of carbon monoxide is dangerous.)

3. One of the major problems facing industrialized nations is water pol-
   lution. Rivers and lakes become polluted with various types of waste
   products which can kill plant and marine life. Once pollution of a river
   is stopped, the river will clean itself fairly rapidly. However, as this ex-
   ample will illustrate, large lakes require much longer to clean themselves
   by the natural process of clean water fl.owing into the lake and polluted
   water fl.owing out of the lake. If CL is the concentration of pollution in
   a lake and V is the volume of the lake, then the total amount of pollu-
   tants in the lake is Q =CL V. If r is the rate at which water enters and
   leaves a lake and if Cin is the concentration of pollutants entering the
   lake, then we have

dQ d(cLV)

dt = dt =rein - rcL = r(Cin - CL)·

If we assume the volume of the lake is constant, then dividing by V, we

find the concentration of pollution in the lake, CL, satisfies the differen-

tial equation

dcL r(Cin - CL)

dt v

Lake Michigan has a volume of USO mi^3 and the yearly fl.ow rate is r =

38 mi^3 /yr. Assuming at time t = 0 the concentration of pollutants in

Lake Michigan is cL(O) = 0.43, assuming the concentration of pollutants

in the entering water is successfully reduced to 0.053, and assuming the

water in the lake is well mixed, how many years will it take to reduce

t he pollution concentration in Lake Michigan to 0.33? 0.253? 0.23?

4. Let the subscript e denote Lake Erie and the subscript o denote Lake

Ontario. The volumes of these lakes are Ve= ll6 mi^3 and V 0 = 393 mi^3.

Pollution enters Lake Onta rio from Lake Erie, from rivers, and from

water run off from surrounding land. Suppose the rate of water entering

Lake Ontario from Lake Erie is r e = 85 mi^3 /yr and the rate of water

entering from non-Erie sources is r = 14 mi^3 /yr. Assuming the rate at

which water leaves Lake Ontario is r 0 =Te +r = 99 mi^3 /yr, the amount

of pollutants in Lake Ontario, Q = c 0 V 0 , satisfy the differential equation

where Ce is the concentration of pollutants in Lake Erie and c is the

concentration of pollutants in non-Erie sources. The first term on the

right represents the amount of pollutants entering Ontario from Erie,