A model for the height of Observatory Hill is given by 1 z = f(x, y) = (1+4x+6xy+12y2)3/2 i) Show that 47 +6ry + 12y can be written as the matrix expression 4x +6ry + 12y = (x, y)S $(3). where S is a symmetric matrix, and obtain an explicit form for S. ii) Obtain the eigenvalues and eigenvectors of S, and hence the diagonal represent S = 0 DO, specifying the elements of D and O. iii) By introducing the new variables ( )= 0 () show that 4x + 6xy+12y = 3u+ 13v. iv) By making a further change of variable, U= cos 3 cos and V= Si 13 show that the volume of Observatory Hill can be written as V v=dz = [_dz_ dyf(x, y) = 5 do 1 de (1 + 03/2 Sot 1 2m and evaluate the integral. sin o