A spherical loaf of bread is cut into parallel slices of equal thickness. Show that, after any number of the slices have been eaten, the area of crust remaining is proportional to the number of slices remaining. A European ruling decrees that a parallel-sliced spherical loaf can only be referred to as crusty if the ratio of volume V (in cubic metres) of bread remaining to area A (in square metres) of crust remaining after any number of slices have been eaten satisfies V < A. Show that the radius of a crusty parallel-sliced spherical loaf must be less than 2 2 3 metres. [The area A and volume V formed by rotating a curve in the xy plane round the x-axis from x = a t to x = a are given by A = 2 a at y ( 1 + (dy dx )2 )1 2 dx , V = a at y 2 dx . ]If y = f (x), the inverse of f is given by Lagranges identity: f 1 (y) = y + 1 1 n! d n1 dy n1 [ y f (y) ]n , when this series converges. (i) Verify Lagranges identity when f (x) = ax. (ii) Show that one root of the equation x 1 4 x 3 = 3 4 is x = 0 3 2n+1 (3n)! n!(2n + 1)! 43n+1 . () (iii) Find a solution for x, as a series in , of the equation x = ex . [You may assume that the series in part (ii) converges and that the series in parts (i) and (iii) converge for suitable a and .]