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4024/22 Oct/Nov 2020, Question 4: Equating Volumes: Melting and Recasting

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 9 marks

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The question, in short

For a solid made from a cylinder (diameter 9 cm, height 16 cm) joined to a hemisphere, work with sphere formulas to find a radius and a surface area, then find a cuboid volume lower bound.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (ai) · 4 marks

Given the hemisphere and cylinder have equal volumes, show the hemisphere radius rounds to 7.86 cm.

  1. 1Volume of cylinder = π × 4.5² × 16 = 324π
  2. 2Volume of hemisphere = 1/2 × 4/3 × π × r³ = 2/3 πr³
  3. 32/3 πr³ = 324π, so r³ = 486
  4. 4r = ∛486 = 7.862… = 7.86 cm (2 d.p.)

Answerr = 7.862...

Part (aii) · 3 marks

Work out the total surface area of the combined solid.

  1. 1Hemisphere radius r = 7.862… cm (from part (i)); cylinder radius = 4.5 cm
  2. 2Curved surface of cylinder = π × 9 × 16 = 144π = 452.38… cm²
  3. 3Curved surface of hemisphere = 2πr² = 2π × 7.862…²
  4. 4Exposed flat ring plus cylinder end = πr² − π × 4.5² + π × 4.5² = πr²
  5. 5Total = 144π + 3π × 7.862…² = 452.38… + 582.55… ≈ 1030 cm²

Answer1030 or 1040 or 1034.6 to 1035.1...

Part (b) · 2 marks

Using 8 cm, 4 cm and 6 cm each rounded to the nearest cm, find the smallest possible cuboid volume.

  1. 1Lower bounds: 7.5 cm, 3.5 cm, 5.5 cm
  2. 2Lower bound of volume = 7.5 × 3.5 × 5.5 = 144.375 cm³

Answer144.375

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