An underwater pressure sensor measures the pressure deviation, PPP, from a baseline at a depth of ddd decametres. The relationship is modeled by the function
P(d)=d4−3d3−5d+12,d>0 P(d) = d^4 - 3d^3 - 5d + 12, \quad d > 0 P(d)=d4−3d3−5d+12,d>0Show that the depth α\alphaα at which the pressure deviation is zero lies in the interval [3,4][3, 4][3,4].
Show that the equation P(d)=0P(d) = 0P(d)=0 can be rewritten as
d=3d2+5−12d3 d = \sqrt[3]{3d^2 + 5 - \frac{12}{d}} d=33d2+5−d12The iterative formula
dn+1=3dn2+5−12dn3 d_{n+1} = \sqrt[3]{3d_n^2 + 5 - \frac{12}{d_n}} dn+1=33dn2+5−dn12is used to find the value of α\alphaα.
Starting with d1=3.2d_1 = 3.2d1=3.2 and using the iterative formula,
(i) find, to 4 decimal places, the value of d2d_2d2.
(ii) find, to 4 decimal places, the value of α\alphaα.
