A large reservoir is contaminated with a chemical pollutant. The concentration, PPP parts per billion (ppb), of the pollutant after ttt days of treatment can be modelled by
P=P0e−kt P = P_0 e^{-kt} P=P0e−ktwhere P0P_0P0 is the initial concentration and kkk is a positive constant. The model is considered accurate for high concentrations.
It takes 22.5 days for the concentration of the pollutant to reach 40% of its initial value.
Determine the number of weeks required for at least 97% of the original pollutant to be removed.
Find the percentage of the initial concentration remaining after 4 weeks. Give your answer to two significant figures.
Explain why the model can only provide an estimate for the actual concentration observed.
Explain why the model is invalid in the very long run as t→∞t \to \inftyt→∞.
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.