本题目来源于试卷: Differential Equations，类别为 IB数学
[问答题]
The population P of fish in a /4x,)m 5d2fqxq*d2jmo yap ap lake aft nut/ls-ist -fx5pgk, d 5e)(ler t weeks can be modelled by the differential l5 i5l -(sgtx- feksndpt ),/uequation.
$\frac{\mathrm{d} P}{\mathrm{~d} t}=k \sqrt{P}, \quad k, t>0$
1. Show that the population of fish is given by
$P(t)=\left(\frac{k t}{2}+\sqrt{P_{0}}\right)^{2}, \quad t>0$
where $ P_{0}$ is the initial fish population.
It is known that the initial fish population was 3000 , and that 24 weeks later the population had doubled in size.
2. Find the value of k to three significant figures.
3. Estimate the number of fish after 30 weeks to the nearest integer.
After a careful adjustment it is found that the model that best describes the fish population is given by
$\frac{\mathrm{d} P_{2}}{\mathrm{~d} t}=(1.89+3 \cos (0.2 \pi t)) \sqrt{P_{2}}$
where t is the time measured in weeks, $t \geq 0 $.
4. Verify that $ P_{2}=\left(\frac{1.89 t}{2}+\frac{30 \sin (0.2 \pi t)}{4 \pi}+\sqrt{3000}\right)^{2} $ is the solution of this new differential equation.
5. Sketch the graph of P_{2}(t) and the graph of P(t) found in parts (a) and (b) on the same axes, for $ 0 \leq t \leq 50 $.
6. Use $ P_{2}(t) $ to estimate the number of whole weeks it takes for the population to reach 5000 fish.
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