Volume of Prisms — Solutions

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1. Volume of rectangular prisms Fluency

   1. 60 cm³
   2. 48 m³
   3. 216 cm³
   4. 100 m³
   5. 84 cm³
   6. 480 mm³
   7. 54 m³
   8. 36 cm³

2. Volume of triangular prisms Fluency

   1. 60 cm³  (A = ½×4×3 = 6; V = 6×10)
   2. 120 m³  (A = ½×6×5 = 15; V = 15×8)
   3. 288 cm³  (A = ½×8×6 = 24; V = 24×12)
   4. 90 mm³  (A = ½×5×4 = 10; V = 10×9)
   5. 140 m³  (A = ½×10×7 = 35; V = 35×4)
   6. 90 cm³  (A = ½×3×3 = 4.5; V = 4.5×20)
   7. 180 cm³  (A = ½×12×5 = 30; V = 30×6)
   8. 90 m³  (A = ½×9×4 = 18; V = 18×5)

3. Volume of other prisms Fluency

   1. 256 cm³  (A = ½×(6+10)×4 = 32; V = 32×8)
   2. 300 m³  (A = ½×(3+7)×5 = 25; V = 25×12)
   3. 140 cm³  (A = 4×2 + 2×3 = 14; V = 14×10)
   4. 630 mm³  (A = ½×(5+9)×6 = 42; V = 42×15)
   5. 120 m³  (A = 6×2 + 3×4 = 24; V = 24×5)
   6. 495 cm³  (A = ½×(8+14)×5 = 55; V = 55×9)
   7. 161 cm³  (A = 5×3 + 4×2 = 23; V = 23×7)
   8. 126 m³  (A = ½×(4+10)×3 = 21; V = 21×6)

4. Find the missing dimension Understanding

   1. h = 5 cm  (120 ÷ (6×4) = 5)
   2. w = 8 m  (360 ÷ (9×5) = 8)
   3. l = 7 cm  (A_base = ½×6×4 = 12; 84 ÷ 12 = 7)
   4. l = 6 mm  (210 ÷ (7×5) = 6)
   5. b = 6 m  (A_base = 180÷10 = 18; ½×b×6=18; b=6)
   6. h = 6 cm  (432 ÷ (12×6) = 6)
   7. h = 4 cm  (A_base = 90÷9 = 10; ½×5×h=10; h=4)
   8. l = w ≈ 4.9 m  (l²×4 = 96; l² = 24; l = √24 ≈ 4.9 m)

5. Volume then convert to litres Understanding

   1. 3000 cm³ = 3 L
   2. 60 000 cm³ = 60 L
   3. 48 000 cm³ = 48 L  (A=½×40×30=600; V=600×80)
   4. 4000 cm³ = 4 L
   5. 288 000 cm³ = 288 L  (120×80×30)
   6. 30 000 cm³ = 30 L
   7. 12 000 cm³ = 12 L  (A=½×24×20=240; V=240×50)
   8. 44 100 cm³ = 44.1 L

6. Problem Solving Problem Solving

   1. 1. 1000 cm³
      2. h = 20 cm  (1000 ÷ (10×5) = 20)
      3. 1 L exactly (since h = 20 cm is already exact)
   2. 1. 1350 cm²  (½×60×45)
      2. 150 cm
      3. 202 500 cm³ = 202.5 L  (1350×150)
   3. 1. 0.36 m³  (3×1.2×0.1)
      2. 360 000 cm³
      3. $64.80  (0.36×$180)

7. Find missing dimension Problem Solving

   1. A_trap = V ÷ l = 840 ÷ 12 = 70 cm². ½×(8+12)×h = 70; 10h = 70; h = 7 cm
   2. A_tri = 270 ÷ 15 = 18 m². ½×9×h = 18; h = 4 m
   3. l = V ÷ (w × h) = 1.5 ÷ (1.5 × 0.5) = 1.5 ÷ 0.75 = 2 m
   4. A_tri = 3.6 ÷ 4 = 0.9 m². ½×1.8×h = 0.9; 0.9h = 0.9; h = 1 m

8. Composite prisms Problem Solving

   1. V₁ = 10×4×3 = 120 cm³; V₂ = 10×8×5 = 400 cm³; Total = 520 cm³
   2. Cross-section: 8×2 + 2×6 = 16 + 12 = 28 cm². V = 28 × 5 = 140 cm³
   3. V_tri = ½×6×4×10 = 120 m³; V_rect = 6×4×10 = 240 m³; Total = 360 m³
   4. Cross-section = 7×3 + 4×2 = 21 + 8 = 29 cm². V = 29 × 8 = 232 cm³

9. Design and cost Problem Solving

   1. 1. 25 × 18 × 12 = 5400 cm³
      2. 5400 cm³ = 5.4 L
      3. 200 ÷ 5.4 ≈ 37.0 → 38 boxes (round up)
   2. 1. V = ½×2.4×1.8×3.5 = 7.56 m³
      2. 7.56 ÷ 2 = 3.78 m³ each. Yes, two people each get 3.78 m³ > 3 m³ minimum.
   3. 1. 60 × 30 × 10 = 18 000 cm³ = 18 L
      2. 300 ÷ 18 = 16.&overline;6 → 17 bags (round up)

10. Multi-step volume problems Problem Solving

    1. V = 50×2.5×2 = 250 m³ = 250 000 L. Time = 250 000 ÷ 5000 = 50 minutes
    2. V_bar = ½×3×2×15 = 45 cm³. 24 bars = 24 × 45 = 1080 cm³. Minimum box volume = 1080 cm³.
    3. 1. 20 × 15 × 6 = 1800 m³
       2. Each pallet footprint = 1.2 × 1 = 1.2 m². Floor = 20 × 15 = 300 m². 300 ÷ 1.2 = 250 pallets
    4. A_trap = ½×(0.8+1.2)×0.5 = 0.5 m². V = 0.5×50 = 25 m³. Time = 25 ÷ 0.2 = 125 seconds ≈ 2 minutes 5 seconds