The mass, M M\,M milligrams, of a substance produced in a chemical reaction is modeled by the equation
M=1200e0.4t5+e0.4tt≥0 M = \frac{1200e^{0.4t}}{5 + e^{0.4t}} \quad t \ge 0 M=5+e0.4t1200e0.4tt≥0where t t\,t is the time in hours after the reaction begins.
Determine the initial mass of the substance produced.
Find the upper limit for the mass of the substance according to this model.
Calculate the time, after the start of the reaction, when the mass reaches 900 mg. Give your answer in hours and minutes to the nearest minute.
Show that
dMdt=Ke0.4t(5+e0.4t)2 \frac{dM}{dt} = \frac{Ke^{0.4t}}{(5 + e^{0.4t})^2} dtdM=(5+e0.4t)2Ke0.4twhere K K\,K is a constant to be determined.
Given that at time t=Tt = Tt=T, the rate of production is dMdt=24\displaystyle \frac{dM}{dt} = 24dtdM=24 mg/h, find the value of T T\,T to one decimal place. (Solutions relying entirely on calculator technology are not acceptable.)
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.