The rate of accumulation of a specific pollutant in a coastal reservoir, RRR, measured in kg/hour, is modeled by the function R(t)=3t3t2+9R(t) = \frac{3t^3}{t^2+9}R(t)=t2+93t3 for 0≤t≤80 \le t \le 80≤t≤8, where ttt represents the time in hours after a filtration system failure.
A researcher approximates the total mass of pollutant accumulated during this 8-hour period, MMM, using the trapezium rule with n=4n = 4n=4 equal intervals. (i) State the number of ordinates that the researcher uses. (ii) Calculate the approximation for MMM that the researcher should obtain. Give your answer correct to two decimal places.
Show that the exact mass of pollutant accumulated, given by M=∫08R(t) dtM = \int_{0}^{8} R(t) \, dtM=∫08R(t)dt, is exactly 96−272ln(739)96 - \frac{27}{2}\ln\left(\frac{73}{9}\right)96−227ln(973). Fully justify your answer.
Explain what would happen to the researcher's approximation in part (a)(ii) as n→∞n \to \inftyn→∞.
Practise AQA A Level Maths 1.12 I: Numerical methods (A-level only) with exam-style questions for A Level Maths. 107 questions covering 1.12.1 Locating roots by change of sign (A-level only), 1.12.2 Iterative methods and Newton-Raphson (A-level only), 1.12.3 Numerical integration (A-level only), and 1.12.4 Numerical methods in context (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.
