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Modeling with Functions · Paper 3 — Free Maths AA HL Practice Questions

1ChallengeLAQReal-world applications of functions (e.g. growth models)s11 marksPaper 3~17 min
A biologist studies the population growth of a rare bird species on an isolated island. The population PP (in hundreds) at time tt (in years) satisfies the logistic differential equation dPdt=15P ⁣(1−PK)\frac{dP}{dt} = \frac{1}{5}P\!\left(1 - \frac{P}{K}\right) where KK is the carrying capacity (in hundreds) and P(0)=1P(0) = 1.
(a)
Show that the general solution satisfying P(0)=1P(0) = 1 is P(t)=K1+(K−1)e−t/5P(t) = \frac{K}{1 + (K-1)e^{-t/5}} [5 marks]
(b)
Given K=10K = 10, calculate the time at which the population reaches 90%90\% of the carrying capacity. [3 marks]
(c)
For the model with K=10K = 10, the growth rate dPdt\frac{dP}{dt} is a function of PP. (i) Determine the value of PP at which dPdt\frac{dP}{dt} is maximised. [1 mark]
(ii) Hence analyse how the rate of population growth changes as PP increases from P=5P = 5 to P=9P = 9, and explain what this implies about the time taken for the population to grow from 50%50\% to 90%90\% of KK compared to the time taken to grow from 10%10\% to 50%50\% of KK. Total: 12 marks [3 marks]
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