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4MA1/2HR Jun 2024, Question 24: Composite Perimeter, Area and Volume

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2HR (Higher) · Calculator · 5 marks

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The question, in short

A solid S combines a cone and hemisphere with equal radius x; total height is 4 times the hemisphere radius. A sphere of radius kx has 12.5 times the volume of S. Find k, then for similar solid T with 512 times volume of S, find d.

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

Part (a) · 4 marks

Work out the value of k for a sphere with radius kx whose volume is 12.5 times the volume of S.

  1. 1Cone height = 4x − x = 3x, so cone volume = ⅓πx² × 3x = πx³
  2. 2Hemisphere volume = ⅔πx³, so volume of S = 5/3 πx³
  3. 3Sphere: 4/3 π(kx)³ = 12.5 × 5/3 πx³, so 4k³ = 62.5
  4. 4k³ = 15.625, so k = ∛15.625 = 2.5

Answer2.5

Part (b) · 1 mark

Solid T is similar to S with volume 512 times S; find d, the factor multiplying S's total surface area.

  1. 1Volume scale factor from S to T is 512
  2. 2Linear scale factor = ∛512 = 8
  3. 3Area scale factor = 8² = 64
  4. 4d = 64

Answer64

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