A biochemist is studying the production of a specific enzyme in a bioreactor. The mass of the enzyme, MMM mg, in the reactor, ttt hours after the reaction begins, is modelled by the equation
M=1200e0.5t5+e0.5tt≥0 M = \frac{1200e^{0.5t}}{5 + e^{0.5t}} \quad t \ge 0 M=5+e0.5t1200e0.5tt≥0Determine the initial mass of the enzyme in the bioreactor.
According to this model, find the limiting value of the enzyme's mass as ttt becomes very large.
Calculate the time elapsed since the start of the reaction when the mass of the enzyme is exactly 900 mg. Give your answer in hours and minutes to the nearest minute.
Show that
dMdt=Ke0.5t(5+e0.5t)2 \frac{dM}{dt} = \frac{Ke^{0.5t}}{(5 + e^{0.5t})^2} dtdM=(5+e0.5t)2Ke0.5twhere KKK is a constant to be determined.
At time t=Tt = Tt=T, the rate of enzyme production is 40 mg/h. Find the value of TTT, giving your answer 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.