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4024/22 Oct/Nov 2022, Question 9: Volume and Surface Area of a Sphere

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

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The question, in short

Use the hemisphere bowl diagram with large diameter 20 cm and rim width 2 cm to work with its surface area, wood mass, and a similar bowl.

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

Part (a) · 3 marks

Show the bowl's total surface area equals 364π cm² using the two hemisphere radii from the 20 cm diameter and 2 cm rim.

  1. 1Outer curved surface = 1/2 × 4π × 10² = 2π × 10² = 200π
  2. 2Inner curved surface (radius 10 − 2 = 8) = 2π × 8² = 128π
  3. 3Rim = π × 10² − π × 8² = 36π
  4. 4Total = 200π + 128π + 36π = 364π cm²

Answer2π × 10² + 2π × 8² + π × 10² − π × 8², completed to 364π with at least one intermediate step

Part (b) · 3 marks

With wood density 0.74 g per cm³, calculate the mass of the bowl.

  1. 1Large radius = 20 ÷ 2 = 10 cm; small radius = 10 − 2 = 8 cm
  2. 2A hemisphere is half a sphere: volume = 2/3 πr³
  3. 3Volume = 2/3 π × 10³ − 2/3 π × 8³ = 1022.06… cm³
  4. 4Mass = 1022.06… × 0.74 = 756.3… ≈ 756 g

Answer756 or 756.2 to 756.4…

Part (c) · 3 marks

For a similar bowl of surface area 546π cm², calculate its mass given the first bowl's mass 756.3 g.

  1. 1First bowl volume = 2/3 π × 10³ − 2/3 π × 8³ = 1022.06… cm³
  2. 2First bowl mass = 1022.06… × 0.74 = 756.3… g (from part b)
  3. 3Area scale factor = 546π ÷ 364π = 1.5, so length factor = √1.5
  4. 4Mass scale factor = (√1.5)³ = 1.837…
  5. 5Mass = 756.3… × 1.837… = 1389.4… ≈ 1390 g

Answer1390 or 1388 to 1390

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Also searched as: 4024/22/O/N/22 · 4024_w22_qp_22 · 4024 w22 22 · question 9.
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