The rate of mass accumulation in a sediment trap, MMM (in mg/year), is modeled by the function:
M(t)=4t3+5t2−12t+15t2+3,t≥0 M(t) = \frac{4t^3 + 5t^2 - 12t + 15}{t^2 + 3}, \quad t \ge 0 M(t)=t2+34t3+5t2−12t+15,t≥0where t t\,t is the time in years since the study began.
Given that
M(t)≡At+B+Ct+Dt2+3 M(t) \equiv At + B + \frac{Ct + D}{t^2 + 3} M(t)≡At+B+t2+3Ct+D(i) Find the values of the constants AAA, B B\,B and CCC.
(ii) Show that D=0D = 0D=0.
Hence, using algebraic integration, find the total mass accumulated between t=1t = 1t=1 and t=3t = 3t=3, giving your answer in the form p+qlnkp + q \ln kp+qlnk, where ppp, q q\,q and k k\,k are integers and k k\,k 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.
