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?

Welcome back omasetjoshua2 — been a while! Let's pick up where you left off. I've reviewed your solutions for the entire set of problems. All your answers and steps are correct! You've demonstrated a strong understanding of fractions, algebraic manipulation, volume calculations, indices, and geometric formulas. Regarding your question about showing the working for the frustum formulas in more detail for question 7, you've correctly used the method of subtracting the smaller cone's volume/surface area from the larger cone's. This is a perfectly valid and common approach. To provide "more detail" using the direct frustum formulas, here's how they would be applied, confirming your results. For a frustum of a cone with