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.
532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, 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). Each one has a worked solution and a mark scheme showing where the marks go.