For some elements such as phosphorus, human activity has strongly perturbed the

Question

For some elements such as phosphorus, human activity has strongly perturbed the natural cycle. Consider the case of a lake which (for simplicity) we will assume contained no dissolved phosphorus initially. Farming moved into the valley and run-off from the use of fertilizer added considerable quantities of P to the lake. This induced the growth of plants that removed dissolved P from lake water. The P is removed in proportion to how much is in the lake. In this case, every year the plants remove 15% of the P that is present. This system can be described by the following equation:

            where P = mass of phosphorus atoms at time t, R = rate of addition of phosphorus (assumed to be 1000 kg/yr), ? = fraction of P removed every year (assumed to be 0.15 yr

Explanation / Answer

first do 2)
take derivative

dP/dt = R/lambda ( lambda e^(-lambda t))

= R e^(-lambda t)

1)

we want
dP/dt/P = 0.01

R e^(-lambda t)/ ( R/lambda (1 -e^(-lambda*t)))

0.15*e^(-0.15*t)/(1 -e^(-0.15*t))=0.01

t=7.39 s

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