The rate of production of a biofuel in a research facility, R(t)R(t)R(t) kilolitres per hour, is modelled by the formula
R(t)=4t3+5t2+8t+15t2+3,1≤t≤3 R(t) = \frac{4t^3 + 5t^2 + 8t + 15}{t^2 + 3}, \quad 1 \le t \le 3 R(t)=t2+34t3+5t2+8t+15,1≤t≤3where ttt is the time in hours from the start of a production cycle. Given that
R(t)≡At+B+Ct+Dt2+3 R(t) \equiv At + B + \frac{Ct + D}{t^2 + 3} R(t)≡At+B+t2+3Ct+D(i) find the values of the constants AAA, BBB and CCC.
(ii) show that D=0D = 0D=0.
Hence, using algebraic integration, calculate the total volume of biofuel produced between t=1t = 1t=1 and t=3t = 3t=3, giving your answer in the form p+qlnkp + q \ln kp+qlnk, where ppp, qqq and kkk are integers and kkk is prime.
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.
