大师作文网A biologist studying fluctuations in the size of a particula
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A biologist studying fluctuations in the size of a particular population decides to investigate a model for which dp/dt=kPcoskt,where P is the size of the population at time t days and k is a positive constant.
(a) Given that p=P₀ when t=0,express P in terms of k,t and P₀
(b)Find the ratio of the maximum size of the population to the minimum size.
(a) Given that p=P₀ when t=0,express P in terms of k,t and P₀
(b)Find the ratio of the maximum size of the population to the minimum size.
dp/dt = kpcoskt
Separate the variables:dp/p = k(coskt)dt
Integrate both sides:lnp = sinkt + c
p = e^(sinkt+c) = Ce^(sinkt)
When t = 0,p = Po,therefore,
Po = Ce^(sin0) = C
p = Poe^(sinkt) --------Answer for Part (a)
Since e^x is a monotonically increasing function,for x1 < x2,e^x1
Separate the variables:dp/p = k(coskt)dt
Integrate both sides:lnp = sinkt + c
p = e^(sinkt+c) = Ce^(sinkt)
When t = 0,p = Po,therefore,
Po = Ce^(sin0) = C
p = Poe^(sinkt) --------Answer for Part (a)
Since e^x is a monotonically increasing function,for x1 < x2,e^x1
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