A civil engineer is modeling the optimal depth ddd, in metres, of a high-capacity drainage channel. The relationship between the depth and the surrounding soil pressure leads to the equation:
d3+50d=800 d^3 + 50d = 800 d3+50d=800Show that the equation has a root α\alphaα in the interval [7,8][7, 8][7,8].
Using the iteration formula
dn+1=800−50dn3withd1=7.5 d_{n+1} = \sqrt[3]{800 - 50d_n} \quad \text{with} \quad d_1 = 7.5 dn+1=3800−50dnwithd1=7.5(i) Calculate the value of d2d_2d2, giving your answer to 4 decimal places. (ii) Determine the value of α\alphaα, corrected to 4 decimal places, through repeated iteration.
