(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

An ecologist studies the birds at Mai Po Nature Reserve. Only 21% of the birds are "residents", i.e. found throughout the year. The remaining birds are migrants. The ecologist suggests that the number N(t) of a certain species of migrants can be modelled by the function

N(t) = 3000 / (1 + ae^{-bt}),

where a, b are positive constants and t is the number of days elapsed since the first one of that species of migrants was found at Mai Po in that year.

(a) This year, the ecologist obtained the following data:

N(5) = 250, N(10) = 870, N(15) = 1940, N(20) = 2670.

(i) Express ln ([tex]\frac{3000}{N(t)}[/tex] - 1) as a linear function of t.

(ii) Use the graph paper below to estimate graphically the values of a and b correct to 1 decimal place.

(b) Basing on previous observations, the migrants of that species start to leave Mai Po when the rate of change of N(t) is equal to one hundredth of N(t). Once they start to leave, the original model will not be valid and no more migrants will arrive. It is known that the migrants will leave at the rate r(s) per day where r(s) = 60 [tex]\sqrt{s}[/tex] and s is the number of days elapsed since they started to leave Mai Po. Using the values of a and b obtained in (a)(ii),

(i) find N'(t), and show that N(t) is increasing;

(ii) find the greatest number of the migrants which can be found at Mai Po this year;

(iii) find the number of days in which the migrants can be found at Mai Po this year.

(Answers

(a)(i) -bt + ln a

(a)(ii) a = 49.4, b = 0.3

(b)(i) 3000 * 49.4 * 0.3 * e^{-0.3t}/ (1 + 49.4e-^{0.3t})^{2}

(b)(ii) 2900

(b)(iii) 42)

2. Relevant equations

Differentiation and Integration Rules

3. The attempt at a solution

I don't know how to solve part (b)(iii).

Is it necessary to solve for s of r(s) = 2900 or [tex]\int[/tex][tex]^{s}_{0}[/tex] r(s) ds = 2900?

Can anyone tell me how to solve it?

Thank you very much!

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# Application of Integration III

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