An environmental scientist is monitoring the breakdown of a pollutant in a controlled bioremediation tank. The concentration of the pollutant PPP parts per million (ppm) at time ttt hours after the treatment begins is modelled by the equation:
P=10(5+λe−kt) P = 10(5 + \lambda e^{-kt}) P=10(5+λe−kt)where λ\lambdaλ and kkk are positive constants.
Initially, the concentration of the pollutant is 200200200 ppm. After 7 hours, the concentration falls to 140140140 ppm.
Find the concentration of the pollutant after 24 hours. Give your answer to three significant figures.
Determine the limiting concentration of the pollutant as t→∞t \to \inftyt→∞, giving a reason for your answer.
Find the time taken, in hours, for the concentration to fall to 555 ppm above the limiting concentration.
Explain why the model might need to be adjusted if the remediation process was conducted in an outdoor reservoir.
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.