Here you go — solutions for this set: Express as a fraction in lowest terms 0.15 Let x = 0.151515... 100x = 15.151515... Subtract: 99x = 15 → x = (15)/(99) = (5)/(33) (ii) 0.125 = (125)/(1000) = (1)/(8) Make r the subject of y = (1 + (z)/(r))3 y1/3 = 1 + (z)/(r) y1/3 − 1 = (z)/(r) r = (z)/(y1/3) - 1 Metal recast into cylinder Volume of block = 5 x 10 x 12 = 600\,m3 Cylinder volume = pi r2 h = 600, where h = 5m pi r2 (5) = 600 → r2 = (600)/(5pi) = (120)/(pi) r = sqrt((120)/(pi)) = sqrt(38.197) = 6.18\,m to 2 d.p. 6(a) If 3 x 9^x−1 = 81^-x, find x Write all as powers of 3: 3 x (32)^x−1 = (34)^-x 31 x 32x−2 = 3⁻⁴x 32x−1 = 3⁻⁴x So 2x - 1 = -4x → 6x = 1 → x = (1)/(6) 6(b) Simplify ((2)/(3))⁻⁴ x ((3)/(5))⁻³ x ((5)/(8))⁻² = ((3)/(2))4 x ((5)/(3))3 x ((8)/(5))2 = (34)/(24) x (53)/(33) x (82)/(52) = (3 x 5 x 64)/(16) = (960)/(16) = 60 6(c) Evaluate ((1)/(27))⁻²/3 + (81)3/4 ((1)/(27))⁻²/3 = (27)2/3 = (33)2/3 = 32 = 9 (81)3/4 = (34)3/4 = 33 = 27 Sum = 9 + 27 = 36 Bucket ABCD from cone DCZ Height of full cone DCZ = 0.84m, DC = 0.42m diameter → radius R = 0.21m Cut-off cone ABZ: ZY = 0.4m, XY = 0.14m diameter → radius r = 0.07m Height of bucket = 0.84 − 0.4 = 0.44m Volume of bucket = Vol large cone - Vol small cone V = (1)/(3)pi R2 H - (1)/(3)pi r2 h V = (1)/(3)pi (0.212 x 0.84 - 0.072 x 0.4) V = (1)/(3)pi (0.037044 − 0.00196) = (1)/(3)pi (0.035084) V = 0.0367\,m3 or 36.7\,L Curved surface area of bucket Slant height large: L = sqrt(0.842 + 0.212) = 0.8658\,m Slant height small: l = sqrt(0.42 + 0.072) = 0.4061\,m CSA = pi R L - pi r l = pi(0.21 x 0.8658 - 0.07 x 0.4061) CSA = pi(0.1818 - 0.0284) = 0.482\,m2 7(c) Pipe cross section Outer radius R = 10mm, inner radius r = 4mm, pi = 3.14 Area of shaded region = pi R2 - pi r2 = pi(102 - 42) = 3.14(100 - 16) = 3.14 x 84 = 263.76\,mm2 Want me to show the working for the frustum formulas in more detail?