Write 1(N−20)(N+80)\frac{1}{(N - 20)(N + 80)}(N−20)(N+80)1 in partial fraction form.
An invasive species of fish is being removed from a large lake to protect the local ecosystem. The population of this species, NNN (measured in hundreds), is modelled by the differential equation
dNdt=−(N−20)(N+80)500 \frac{dN}{dt} = -\frac{(N - 20)(N + 80)}{500} dtdN=−500(N−20)(N+80)where ttt is the time, in years, from when the removal program began.
Given that the initial population of the species was 12,00012,00012,000 fish (so N=120N = 120N=120 at t=0t = 0t=0),
solve the differential equation to show that
N=40+80e−0.2t2−e−0.2t N = \frac{40 + 80e^{-0.2t}}{2 - e^{-0.2t}} N=2−e−0.2t40+80e−0.2tHence find the time taken for the population of the species to fall to 4,5004,5004,500 fish.
(Solutions relying entirely on calculator technology are not acceptable.)
According to the model, the population will eventually fall to 100k100k100k fish.
State the value of the constant kkk.
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.