A chemical reactor's rate of gas production, R(t)R(t)R(t) in m3/hr\text{m}^3/\text{hr}m3/hr, is monitored over a period of time t t\,t in hours. For the interval 1≤t≤31 \le t \le 31≤t≤3, the production rate is modeled by the function:
R(t)=(6−2t)lnt R(t) = (6 - 2t) \ln t R(t)=(6−2t)lntA graph of this function shows that the production rate starts at zero at t=1t = 1t=1, reaches a maximum, and returns to zero at t=3t = 3t=3.
Use the trapezium rule with 5 ordinates to find an estimate for the total volume of gas produced between t=1t = 1t=1 and t=3t = 3t=3. Give your answer correct to three significant figures.
Use integration to find the exact volume of gas produced and show that it is given by
9ln3−8 9 \ln 3 - 8 9ln3−8Fully justify your answer.
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.
