Topic Review — Surface Area and Volume — Solutions

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1. Surface area — fluency

   1. SA = 2(9×4 + 9×5 + 4×5) = 2(36+45+20) = 2(101) = 202 cm²
   2. SA = 2π(6)² + 2π(6)(9) = 72π + 108π = 180π cm²
   3. r = 8 mm. SA = 2π(8)² + 2π(8)(25) = 128π + 400π = 528π ≈ 1658.8 mm²
   4. 216 = 6s², s² = 36, s = 6 cm
   5. Triangle area = ½×5×12 = 30 cm². Rect. faces: 5×20=100, 12×20=240, 13×20=260. SA = 2(30)+100+240+260 = 60+600 = 660 cm²

2. Volume — fluency

   1. V = π(3.5)²(10) = 122.5π ≈ 384.85 cm³
   2. V = 15×8×4 = 480 cm³
   3. r = 1.2 m. V = π(1.2)²(1.8) = 2.592π ≈ 8.143 m³ = 8143 litres
   4. Area = ½×7×4 = 14 m². V = 14×9 = 126 m³
   5. 500π = πr²(20). r² = 500/20 = 25. r = 5 cm

3. Composite solids and hollow shapes

   1. V_cube = 4³ = 64 cm³. V_cylinder = π(2)²(5) = 20π ≈ 62.83 cm³. Total = 64 + 20π ≈ 126.83 cm³
   2. V_outer = 10×10×30 = 3000 cm³. V_inner = 8×8×30 = 1920 cm³. V_material = 1080 cm³
   3. Length = 120 cm. Outer curved SA = 2π(6)(120) = 1440π ≈ 4523.89 cm²
   4. V = 20×10×1.5 = 300 m³ = 300 000 L. Time = 300 000 ÷ 1500 = 200 minutes = 3 hours 20 minutes

4. Real-world problem solving

   1. Area to paint = 2π(1.2)(2) + π(1.2)² = 4.8π + 1.44π = 6.24π ≈ 19.60 m². Litres = 19.60/8 = 2.45 → 3 litres. Cost = 3×$22 = $66
   2. r = 0.25 m, h = 3.5 m. V = π(0.25)²(3.5) = 0.21875π ≈ 0.687 m³. Cost = 0.687×$280 ≈ $192.44
   3. SA = 2(60×45+60×40+45×40) = 2(2700+2400+1800) = 13 800 cm² = 1.38 m². With 20% extra: 1.656 m². Cost = 1.656×$3.20 = $5.30
   4. SA = 2π(3.5)² + 2π(3.5)(12) = 24.5π + 84π = 108.5π ≈ 340.84 cm². Cost/can = 340.84×0.005 ≈ $1.70. Total = 5000×$1.70 = $8 521.10

5. Mixed extended response

   1. Water trough (triangular prism, base 0.8 m, height 0.5 m, length 4 m):
      1. Area = ½×0.8×0.5 = 0.2 m². V = 0.2×4 = 0.8 m³ = 800 litres
      2. Half full = 800/2 = 400 litres
      3. Two triangular ends: slant sides — isosceles base 0.8 m, height 0.5 m, so slant = √(0.4²+0.5²) = √0.41 ≈ 0.640 m each. Triangle area = 0.2 m². Inner SA = 2(0.2) + 0.8×4 + 2(0.640×4) = 0.4 + 3.2 + 5.12 = 8.72 m²
   2. Silos:
      1. Silo A: V = π(4)²(10) = 160π ≈ 502.65 m³. Silo B: V = π(5)²(6.4) = 160π ≈ 502.65 m³. Both hold the same volume (160π m³).
      2. Silo A curved: 2π(4)(10) = 80π ≈ 251.33 m². Cost = $3 769.91. Silo B curved: 2π(5)(6.4) = 64π ≈ 201.06 m². Cost = $3 015.93. Silo B is cheaper by $753.98.
      3. Silo A 60% full: 0.6×160π×1000 ≈ 301 593 L. Silo B 75% full: 0.75×160π×1000 ≈ 376 991 L. Total ≈ 678 584 litres