Application of Integration III

[chrisyuen]

Homework Statement

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 modeled 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)

Homework Equations

Differentiation and Integration Rules

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!

 

[HallsofIvy]

?? I can find no "part (c)" in what you have written.

 

[chrisyuen]

?? I can find no "part (c)" in what you have written.

Sorry, it should be part (b)(iii) instead of part (c).

 

[HallsofIvy]

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).

So solve N'(t)= N(t)/100 to find T₁, the number of days until they start to leave. Also find N(T₁) to determine how many there are in Mai Po on that date.

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 and s is the number of days elapsed since they started to leave Mai Po.

r(s) is a constant, 60, so the number that leave in s days will be 60s. Solve 60s= N(T₁) to get s= T₂, the number of days from the day the birds start to leave until there are none left.

 

[chrisyuen]

So solve N'(t)= N(t)/100 to find T₁, the number of days until they start to leave. Also find N(T₁) to determine how many there are in Mai Po on that date.


r(s) is a constant, 60, so the number that leave in s days will be 60s. Solve 60s= N(T₁) to get s= T₂, the number of days from the day the birds start to leave until there are none left.

r(s) = 60 [tex]\sqrt{s}[/tex] (but not a constant 60)

Should I need to set the equation: 60 s [tex]\sqrt{s}[/tex] = 2900 to solve this problem?