22   8. (a) Given zn + 1 zn = 2 cos nθ      n ∈  show that 32 cos6 θ ≡ cos 6θ + 6 cos 4θ + 15 cos 2θ + 10 (5) y x O x = –4 x = 4 R Figure 1 Figure 2 Figure 1 shows a solid paperweight with a flat base. Figure 2 shows the curve with equation y = H cos3  x 4              –4  x  4 where H is a positive constant and x is in radians. The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = –4 , the line with equation x = 4 and the x-axis. The paperweight is modelled by the solid of revolution formed when R is rotated 180° about the x-axis. Given that the maximum height of the paperweight is 2 cm, (b) write down the value of H. (1) (c) Using algebraic integration and the result in part (a), determine, in cm3 , the volume of the paperweight, according to the model. Give your answer to 2 decimal places. [Solutions based entirely on calculator technology are not acceptable.] (5) (d) State a limitation of the model. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 23   Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________