Problem Solving with Volume — Solutions

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1. Composite volume (add) Fluency

   1. 156 cm³  (96 + 60)
   2. 168 m³  (120 + 48)
   3. 147 cm³  (105 + 42)
   4. 168 m³  (96 + 72)
   5. 240 cm³  (180 + 60)
   6. 210 m³  (90 + 120)
   7. 168 cm³  (144 + 24)
   8. 500 cm³  (400 + 100)

2. Volume with section removed Fluency

   1. 340 cm³  (400 − 60)
   2. 264 m³  (288 − 24)
   3. 780 cm³  (900 − 120)
   4. 480 cm³  (512 − 32)
   5. 880 m³  (1000 − 120)
   6. 432 cm³  (486 − 54)
   7. 1840 cm³  (2400 − 160... wait: 25×12×8=2400; 5×4×8=160; 2400−160=2240)
   8. 744 m³  (840 − 96)

3. Mixed volume types Fluency

   1. 308 cm³
   2. 180 cm³  (½×8×5×9)
   3. 360 m³
   4. 210 m³  (½×(4+8)×5×7=210)
   5. 420 mm³  (½×12×7×10)
   6. 84 cm³
   7. 297 m³  (½×9×6×11)
   8. 160 cm³  (½×(3+7)×4×8)

4. Compare containers Understanding

   1. A = 240 cm³; B = 240 cm³; Same volume
   2. A = 120 m³ (½×6×4×10); B = 120 m³; Same volume
   3. A = 288 cm³; B = 360 cm³ (½×10×8×9); B holds more
   4. A = 3000 cm³; B = 3000 cm³; Same volume
   5. A = 729 cm³; B = 672 cm³; A holds more
   6. A = 168 m³ (½×(5+9)×4×6); B = 240 m³; B holds more
   7. A = 60 000 cm³; B = 63 000 cm³; B holds more
   8. A = 480 cm³ (½×8×6×20); B = 480 cm³; Same volume

5. Multi-step real-world Understanding

   1. 72 000 cm³ = 72 L; 72 ÷ 2 = 36 minutes
   2. Container = 1.2m×0.8m×0.5m = 0.48 m³ = 480 000 cm³; Each box = 2000 cm³; 480 000 ÷ 2000 = 240 boxes
   3. Full pool = 25×10×1.8 = 450 m³ = 450 000 L; Half full = 225 000 L needed
   4. V = ½×50×30×200 = 150 000 cm³ = 150 L; 150 ÷ 5 = 30 minutes
   5. Box = 30×20×15 = 9000 cm³; Each cube = 5³ = 125 cm³; 9000 ÷ 125 = 72 cubes
   6. Tank = 1.5×1×2 = 3 m³ = 3000 L; 1200 ÷ 3000 = 40%
   7. V = ½×3×0.4×5 = 3 m³; Cost = 3×$220 = $660
   8. V = 180×90×30 = 486 000 cm³ = 486 L; 486 ÷ 9 = 54 minutes = 54 min 0 sec

6. Problem Solving Problem Solving

   1. 1. 3000 cm³
      2. h = 10 cm  (3000 ÷ (20×15))
      3. Base: 300 cm²; 2 long sides: 2×20×10=400 cm²; 2 short sides: 2×15×10=300 cm²; Total = 1000 cm²
   2. 1. Each step = 1.8×0.3×0.18 = 0.0972 m³
      2. Total = 2×0.0972 = 0.1944 m³
      3. 0.1944 ÷ 0.2 = 0.972 → round up to 1 load
   3. 1. 3600 cm³  (15×12×20)
      2. 360 cm³  (15×12×2 — water rises 2 cm across the full base)
      3. 360 cm³  (volume displaced = volume of rock)

7. Pools and tanks Problem Solving

   1. 1. 8 × 4 × 1.2 = 38.4 m³
      2. 38 400 L
      3. 40% needed = 0.4 × 38 400 = 15 360 L
      4. 15 360 ÷ 200 = 76.8 minutes
   2. V = 1×1×2 = 2 m³ = 2000 L. Tank is ¼ full = 500 L; needs 1500 more. 1500 ÷ 5 = 300 minutes
   3. 1. A_trap = ½×(1.5+2.5)×0.6 = 1.2 m². V = 1.2 × 3 = 3.6 m³
      2. 3600 L

8. Storage and packaging Problem Solving

   1. Shelf = 180×60×150 = 1 620 000 cm³. Box = 30×20×15 = 9000 cm³. 1 620 000 ÷ 9000 = 180 boxes
   2. Crate = 120×80×60 = 576 000 cm³. Box = 20×10×15 = 3000 cm³. 576 000 ÷ 3000 = 192 boxes
   3. V = 2.4 L = 2400 cm³. h = 2400 ÷ (20×10) = 12 cm
   4. Box = 50×40×35 = 70 000 cm³. 70 000 ÷ 700 = 100 books (estimate)

9. Composite solid problems Problem Solving

   1. 1. Lower = 1.5×0.4×0.2 = 0.12 m³. Upper = 1.5×0.3×0.2 = 0.09 m³
      2. Total = 0.12 + 0.09 = 0.21 m³
      3. Cost = 0.21 × $180 = $37.80
   2. Block = 4×3×10 = 120 m³. Tunnel = 2×1.5×10 = 30 m³. Remaining = 120 − 30 = 90 m³
   3. Cross-section = 4×0.4 + 2×0.4 = 1.6 + 0.8 = 2.4 m². V = 2.4 × 1.5 = 3.6 m³
   4. Block = 10×8×6 = 480 cm³. Each hole = 2×2×6 = 24 cm³. Two holes = 48 cm³. Solid = 480 − 48 = 432 cm³

10. Extended volume investigations Problem Solving

    1. 1. A_trap = ½×(20+40)×5 = 150 m². V = 150×100 = 15 000 m³
       2. 15 000 kL
       3. 15 000 ÷ 500 = 30 hours
    2. 1. A_trap = ½×(1+3)×20 = 40 m². V = 40×10 = 400 m³
       2. 400 000 L
       3. 400 000 ÷ 3000 = 133.&overline{3} minutes = 2 hours 13 minutes 20 seconds
    3. 1. Outer dimensions = (6 + 1) m × (4 + 1) m = 7 m × 5 m
       2. Path area = 7×5 − 6×4 = 35 − 24 = 11 m². V = 11 × 0.1 = 1.1 m³
       3. Cost = 1.1 × $220 = $242