Part of the water flowing into Lake A is from Lake B Suppose

Part of the water flowing into Lake A is from Lake B. Suppose that the pollution of these lakes ceased suddenly at this year when the amount of pollution in Lake A and B are a₀and b₀respectively. Let a_nand b_nbe the amount of pollution in Lake A and B respectively after year n. Each year 39% of the water in Lake B flows into Lake A and is replaced by rain and pure water flowing in from other sources; also each year 15% of Lake A

Solution

since in lake B every year 39% water is replaced by pure water,so in other words 39% pollution is removed so

we can wirte it as following equation

b(n)=b(n-1)-0.39b(n-1)

it means that whatever is the pollution at the end of any year next year 0.39 percent of that pollution is removed from the pollution that is exist at the end of the year.

so

b(n) =0.61b(n-1)

similary for lake A 39% water comes from lake B,so amount of pollution that reached in lake A from lake B is

0.39b(n-1)

and 15% is removed so 0.15% of pollution is removed first then 0.39% of lake B pollution is added

so for lake A we have following equation.

a(n) = a(n-1)-0.15a(n-1)+0.39b(n-1)-0.39a(n-1)

0.39a(n-1) is subtracted because this much removes from lake A as this much water is replaced.

a(n) = 0.46a(n-1)+0.39b(n-1)

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b.

b(n) =0.61b(n-1)

b(0)=1

b(1)=0.61

b(2)=0.61*0.61=0.37

b(3)=0.61*0.37=0.23

b(4)=0.61*0.23=0.14

b(5)=0.61*0.14=0.08

a(n) = 0.46a(n-1)+0.39b(n-1)

a(1)=0.46+0.39=0.85

a(2)=0.46*0.85+0.39*0.61=0.63

a(3)=0.46*0.63+0.39*0.37=0.43

a(4)=0.46*0.43+0.39*0.23=0.29

a(5)=0.46*0.29+0.39*0.14=0.19