An environmental engineer is monitoring the reduction of a hazardous chemical in a decontamination tank. The concentration of the chemical, C C\,C ppm (parts per million), at time t t\,t hours after the process starts is modeled by the equation
C=C0e−kt C = C_0 e^{-kt} C=C0e−ktwhere C0 C_0\,C0 is the initial concentration and k k\,k is a constant related to the efficiency of the filtration system.
Experiments show that the time taken for the concentration of this chemical to reduce to half its previous value is 3.2 hours.
At 8:00 am, the initial concentration of the chemical in the tank is measured at 800 ppm.
Determine the estimated concentration of the chemical in the tank at 2:00 pm on the same day.
The tank requires a maintenance injection that will increase the concentration by exactly 600 ppm. Safety protocols dictate that the total concentration must never exceed 750 ppm. Find the earliest time that the maintenance injection can be administered. Give your answer to the nearest minute.
State one reason why the concentration predicted by this model might differ from actual measurements taken from the tank.
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.