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.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.