The pressure PPP (in kilopascals) at a specific depth in a hydraulic system is modelled by the equation
g(P)=P5+12P−8000 g(P) = P^5 + 12P - 8000 g(P)=P5+12P−8000Show that the equation g(P)=0g(P) = 0g(P)=0 has a root β\betaβ in the interval [6,7][6, 7][6,7].
Using the iterative formula
Pn+1=8000−12Pn5withP1=6.2 P_{n+1} = \sqrt[5]{8000 - 12P_n} \quad \text{with} \quad P_1 = 6.2 Pn+1=58000−12PnwithP1=6.2(i) Calculate the value of P2P_2P2, giving your answer to 4 decimal places. (ii) By performing repeated iterations, find the value of β\betaβ to 4 decimal places.
