Everyday at noon, a factory dumps 10 tons of pollutants into a lake. Every 24 ho

Question

Everyday at noon, a factory dumps 10 tons of pollutants into a lake. Every 24 hours, one third of all pollutants in the lake are removed by natural processes. Assume the lake starts off pollutant-free.

a) Express the long-run quantity of pollutants in the lake right after a dumpas a geometric series

b) Find the sum of the geometric series found in part a

c) If environmental regulation prohibits more than 15 tons of pollution in the lake at any time, what is the maximum amounts of pollutants the factory can dump into the lake daily?

Explanation / Answer

a) Let at t= 0 , 10 tons of pollutant are dumped

at t= 24 , pollutants remaining = (2/3)*10 + 10 (newly dumped) = 10*( 1+ 2/3)

at t = 48 , pollutants remaining = (2/3)*10*(1+2/3) + 10 = 10*( 1+2/3 +(2/3)^2)

this goes on

so after a long time the quantity of pollutants remaining is equal to

P = 10* ( 1 + 2/3 + (2/3)^2 + (2/3)^3 +...........so on

b) sum of the series = 10 * { 1 / (1-2/3)} = 30 tons

c) if max limit = 15 tons , let maximum dumping daily = x

so we have x *{ 1/(1-2/3)} = 15

=> x = 5 tons

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