10 a An initially clean pond that contains 10000 m 3 of water maintains a

10 a an initially clean pond that contains 10000 m 3

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10. a. An initially clean pond that contains 10,000 m 3 of water maintains a constant volume. The stream flows in at a rate of 200 m 3 /day with 10 μ g/m 3 of phosphate (from fertilizer) in the stream. The pond is well-mixed, and a stream flows out at the same rate. Write a differential equation that describes the concentration of phosphate c ( t ) in the lake, then solve this equation. b. Algae grows well on phosphate. The rate of growth of algae is proportional to the concen- tration of phosphate and the population of algae A ( t ) to the 2/3 power, dA dt = 0 . 05 c ( t ) A 2 / 3 , A (0) = 1000 . Find the population of algae at any time t . 11. Studies of Lake Apopka in Florida show that the alligators there have been exposed to high levels of various estrogen-simulating pesticides, like DDT (or its breakdown DDE), dieldrin, and toxaphenes. Apparently, this exposure has resulted in dramatic decrease in the size of the male alligator penises, which in turn finally spurred our Congress into action. (They were unconcerned when it was shown to have adverse effects on female animal populations.) The levels of these
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estrogenic pesticides in this lake far exceed the safe levels established by the EPA. The effect on penis development is clearly related to the amount of exposure of the embryo, which in turn reflects the amount in the body of the female alligators. a. The growth (weight) of a female alligator can be approximated by the following differential equation: dw dt = 0 . 2(80 - w ) , w (0) = 0 , where w is the weight in kg and t is the time in years. Solve this differential equation. Use the results to determine how long it takes to produce a mature 40 kg alligator. b. The pesticides accumulate (especially in the fatty tissues) as the alligator grows and is not removed. Assume that the intake of pesticides is proportional to the weight of the alligator, so satisfies the differential equation dP dt = kw ( t ) , P (0) = 0 , with k = 600 ( μ g/kg-yr) and P being the μ g of pesticides in an alligator at Lake Apopka. Solve this differential equation. Find the amount of pesticide in an alligator that is 5 years old. c. The concentration, c ( t ) (in μ g/g or ppm), is found by computing c ( t ) = P ( t ) 1000 w ( t ) . Find the concentration in a 5 year old alligator. (Note that officials get concerned when pesticide levels reach 0.1 ppm in an animal.) 12. a. A new pesticide is introduced to a particular region, where a stream becomes contaminated with the pesticide and flows into a 10 6 m 3 lake. The stream flows at a rate of 4000 m 3 /day with a concentration of 15 ng/m 3 . Assuming that the lake is well-mixed and maintains a constant volume, then find the differential equation describing the concentration of this new pesticide in the lake. Solve this differential equation and find the limiting concentration in the lake. b. More realistically, the flow of the stream is seasonal and fluctuates over the year by about 40%. The lake still maintains an almost constant volume, but a better model for the concentration of the pesticide in the lake is given by: dc dt = - 0 . 001(4 - cos(0 . 0172 t ))( c - 15) , with c (0) = 0 .
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