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.
302 exam-style questions on WJEC A Level Maths 1.2 Algebra and Functions, covering 1.2.1 Algebra and Functions, 1.2.2 Algebra and Functions, 1.2.3 Algebra and Functions, 1.2.4 Algebra and Functions, 1.2.5 Algebra and Functions, 1.2.6 Algebra and Functions, 1.2.7 Algebra and Functions, 1.2.8 Algebra and Functions, 1.2.9 Algebra and Functions, 1.2.10 Algebra and Functions, 1.2.11 Algebra and Functions, and 1.2.12 Algebra and Functions. Each one has a worked solution and a mark scheme showing where the marks go.