The concentration of a bio-pollutant in a nature reserve's lake is modeled by
C=12+75(t20)3−15(t20)4 C = 12 + 75\left(\frac{t}{20}\right)^3 - 15\left(\frac{t}{20}\right)^4 C=12+75(20t)3−15(20t)4where CCC is the concentration in parts per billion (ppb) and ttt is the number of years since 1 January 2010.
The model predicts that the pollutant concentration will eventually return to zero at a time TTT. Show that TTT satisfies the equation
T=100T2+128 000T3 T = \sqrt[3]{100T^2 + \frac{128\,000}{T}} T=3100T2+T128000Use the iterative formula Tn+1=100Tn2+128 000Tn3T_{n+1} = \sqrt[3]{100T_n^2 + \frac{128\,000}{T_n}}Tn+1=3100Tn2+Tn128000, with T0=50T_0 = 50T0=50, to find the values of T1,T2,T_1, T_2,T1,T2, and T3T_3T3 to three decimal places.
Explain the relevance of using T0=50T_0 = 50T0=50 in terms of the date.
A neutralizing agent is introduced, and its effectiveness in reducing current pollutant levels is modeled by the supply function
S=6×1.08t S = 6 \times 1.08^t S=6×1.08tUse the models to show that the concentration CCC and the neutralizing supply SSS will be equal during the year 2075.
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.
