The rate of accumulation of a specific enzyme in a bioreactor, R(t)R(t)R(t) in mmol/L per hour, is modeled by the function:
R(t)=7t−2(t+3)2,t≥0 R(t) = \frac{7t - 2}{(t + 3)^2}, \quad t \ge 0 R(t)=(t+3)27t−2,t≥0where t t\,t is the time in hours since the start of the reaction.
Express R(t)R(t)R(t) in the form At+3+B(t+3)2\displaystyle \frac{A}{t + 3} + \frac{B}{(t + 3)^2}t+3A+(t+3)2B, where A A\,A and B B\,B are constants to be found.
The total accumulation of the enzyme, KKK, between t=1t = 1t=1 and t=5t = 5t=5 hours is given by K=∫15R(t) dtK = \int_{1}^{5} R(t) \, dtK=∫15R(t)dt.
Show that K=p+lnqK = p + \ln qK=p+lnq, where p p\,p and q q\,q are rational numbers to be determined.
Practise AQA A Level Maths 1.5 B: Algebra and functions with exam-style questions for A Level Maths. 358 questions covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (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.
