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.)
Practise AQA A Level Maths 1.10 G: Differentiation with exam-style questions for A Level Maths. 321 questions covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
