Volume of Cylinders and Composite Solids — Solutions

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1. Volume of cylinders

   1. V = π(5)²(8) = 200π ≈ 628.3 cm³
   2. V = π(3)²(10) = 90π ≈ 282.7 m³
   3. V = π(6.5)²(4) = 169π ≈ 531.0 cm³
   4. r = 7 mm. V = π(7)²(20) = 980π ≈ 3079.0 mm³
   5. r = 0.6 m. V = π(0.6)²(3.5) = 1.26π ≈ 3.958 m³ ≈ 4.0 m³

2. Volume of prisms

   1. V = 9 × 4 × 6 = 216 cm³
   2. V = 2.5 × 1.8 × 1.2 = 5.4 m³
   3. Triangle area = ½×8×5 = 20 cm². V = 20×12 = 240 cm³
   4. Triangle area = ½×10×6 = 30 m². V = 30×4 = 120 m³

3. Finding missing dimensions from volume

   1. 200π = π(5)²h = 25πh. h = 200/25 = 8 cm
   2. 1000 = πr²(20). r² = 1000/(20π) = 15.915…. r = √15.915 ≈ 3.99 cm
   3. 360 = 10 × 6 × h = 60h. h = 6 cm
   4. 180 = (½×9×5) × L = 22.5L. L = 180/22.5 = 8 m

4. Unit conversion and capacity

   1. V = 80×60×50 = 240 000 cm³. Capacity = 240 000 ÷ 1000 = 240 litres
   2. V = π(1.5)²(2.8) = 6.3π ≈ 19.792 m³. Capacity = 19.792 × 1000 ≈ 19 792 litres
   3. 250 000 L ÷ 1000 = 250 m³
   4. 3 L = 3000 mL. 3000 ÷ 250 = 12 glasses

5. Composite solid (rectangular prism + triangular prism on top)

   1. V = 10 × 6 × 4 = 240 cm³
   2. Triangle area = ½×6×3 = 9 cm². V = 9 × 10 = 90 cm³
   3. Total V = 240 + 90 = 330 cm³
   4. Sketch: rectangular prism (10×6×4) with a triangular prism (10×6 base, triangle height 3) sitting on top, forming a house shape.

6. Hollow cylindrical pipe

   1. Length = 2 m = 200 cm. V_outer = π(5)²(200) = 5000π ≈ 15 707.96 cm³
   2. V_inner = π(4)²(200) = 3200π ≈ 10 053.10 cm³
   3. V_wall = 5000π − 3200π = 1800π ≈ 5654.87 cm³
   4. Mass = 5654.87 × 7.8 = 44 108 g ≈ 44.1 kg

7. Cylindrical fuel tank

   1. r = 0.7 m. V = π(0.7)²(3) = 1.47π ≈ 4.618 m³
   2. 4.618 × 1000 = 4618 litres
   3. Value = 4618 × $2.18 ≈ $10 067
   4. Currently 40% full: 0.4 × 4618 = 1847.2 L. Needed = 4618 − 1847.2 ≈ 2771 litres

8. Swimming pool

   1. V = 15 × 8 × 1.2 = 144 m³
   2. V = 5 × 8 × 2.4 = 96 m³
   3. Total V = 144 + 96 = 240 m³
   4. 240 × 1000 = 240 000 litres
   5. Time = 240 000 ÷ 800 = 300 minutes = 5 hours

9. Drilled rectangular block

   1. V = 20 × 12 × 8 = 1920 cm³
   2. V = π(3)²(8) = 72π ≈ 226.19 cm³
   3. V_remaining = 1920 − 72π ≈ 1920 − 226.19 = 1693.81 cm³
   4. Mass = 1693.81 × 0.6 ≈ 1016.3 g

10. Rainwater tank

    1. V = π(0.9)²(1.8) = 1.458π ≈ 4.580 m³
    2. 4.580 × 1000 = 4580 litres
    3. 4580 ÷ 120 = 38.17, so 38 full days
    4. 850 ÷ 4580 × 100 ≈ 18.6%
    5. 4580 − 1200 = 3380 litres