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.
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.