An orchard contains 1200 apple trees. A farmer observes a fungal infection spreading among the trees. Initially, 40 trees are infected. The number of infected trees is increasing by 25% each day.
The total number of infected trees, xxx, is modelled by
x=A×Bt x = A \times B^t x=A×Btwhere A A\,A and B B\,B are constants and t t\,t is the number of days after the farmer first noticed the infection.
(i) Find the total number of infected trees 6 days after the farmer first noticed the infection based on this model.
(ii) Explain why this model is not realistic in the long term for the orchard.
A refined model assumes the rate of increase of the number of infected trees is given by
dxdt=x(1200−x)4800 \frac{dx}{dt} = \frac{x(1200 - x)}{4800} dtdx=4800x(1200−x)(i) Show that
∫(Cx+D1200−x)dx=∫dt \int \left( \frac{C}{x} + \frac{D}{1200 - x} \right) dx = \int dt ∫(xC+1200−xD)dx=∫dtwhere C C\,C and D D\,D are positive integers to be found.
(ii) Hence, find t t\,t in terms of xxx.
(iii) Find the number of days it takes from when the infection is first noticed until half of the trees are infected.
864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.