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.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
