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Rates of change and differential equations — Question 152

Original QCE Vault practice · 8 marks

Q152 · Practice questionTechnology-activeComplex unfamiliar8 marks

QUESTION 152 (8 marks)

A population is modelled as a continuous quantity N(t)N(t), where tt is measured in days. It satisfies
dNdt=kN(K−N),k>0,K>0.\frac{dN}{dt}=kN(K-N),\qquad k>0,\quad K>0.
The graph shows the growth rate as a function of population. The axis intercepts and the labelled point P are exact. Initially, N(0)=50N(0)=50.
Exact growth-rate curve with intercepts N=0 and N=400 and labelled point P=(100,30).
a)
Use the graph to determine K and k.
[2 marks]
b)
Determine the first time after t=0t=0 when the growth rate returns to its initial value. Give an exact expression and an answer in days, correct to two decimal places.
[6 marks]
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