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.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.