The Lake Cachuma reservoir system is actually more complex than we let on earlie

Question

The Lake Cachuma reservoir system is actually more complex than we let on earlier. Really, run-off water flows mostly into the Gibraltar Reservoir, water from Gibraltar Reservoir flows into Lake Cachuma, and we take water from Lake Cachuma for drinking water. Instead of tracking nitrates, we will track mercury in this problem. Suppose pure water flows into Gibraltar at a rate of rg = 1 trillion L/mo; water flows from Gibraltar into Cachuma at a rate of rc = 1 trillion L/mo; and we drain Cachuma at a rate of rd = 1 trillion L/mo. Each lake has a volume of 10 trillion L. Currently neither lake has any mercury at all, but authorities want to plan for the possibility of a rockslide at the Sunbird Quicksilver Mine, which could deposit up to 10kg of mercury directly into Gibraltar. If this happens, what would be the maximum concentration of mercury that occurs in Cachuma?

Explanation / Answer

suppose in case of a rockslide 10 kg of mercury gets into Gibralter so

concentration at that time = 10kg/10 trillionL = 1kg/trillion L

since flow of fresh water into Gibralter is 1 trillion L/mo so initial flow from Gibralter to Cachuma

is 1kg/mo. Since rate of flow of water from Gibralter to Cachuma is 1 trillion L/mo

and volume of Cachuma is 10 trillion and (1/10)of 10 trillion i.e 1 trillion is drained in first month so

drained water from Cachuma first month has (1/10) kg of mercury.

Weight of mercury added in cachuma in 1st month = (1-1/10) kg = 9/10 kg

(a factor of 1/10 is used since every month 10% is drained from Cachuma)

Corresponding mercury weight in drained water i s = 1/10 kg

Weight of mercury added in cachuma in 2nd month = previous*(1-1/10) = 9/10 kg * (1-1/10) = 81/100 kg

Corresondingly mercury weight in drained water is 9/10 *(1/10) =9/100 kg

Weight of mercury added in cachuma in 3st month = previous*(1-1/10) = 81/100*(1-1/10) = 729/1000 kg

Corresondingly drained water in 3rd month mercury weight is 81/100 *(1/10) = 81/1000

and so on after very long time mercury in Gibraltar tends to zero

so total mercury in Cauchuma's water = 9/10 + (81/100) + (729/1000) + ...

so the first term of infinite g.p is 9/10 and ratio is also 9/10

sum of G.P = first term /(1- ratio) ratio between successive terms here is 9/10 ; first term also 9/10

so sum is (9/10)/ (1-9/10) = 9kg

since volume of lake is 10 trillion liters

so concentration in kg/ trillion Litres = 9/10 (kg/trillion litres)

Corresponding drained water contains (10-9) = 1kg with mercury

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