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4024/11 Oct/Nov 2022, Question 23: Volume and Surface Area of a Cone

Cambridge O Level Mathematics 4024 · Paper 11 (Non-calculator) · Non-calculator · 7 marks

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The question, in short

A cone has radius y cm and slant height 25% larger than its radius. A sphere of radius R cm has surface area equal to the cone's total surface area. Show y = 4R/3, then find the cone's volume in terms of R.

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Part (a) · 3 marks

Using the cone's total surface area (curved plus base) equal to the sphere's surface area 4πR², show that y = 4R/3.

  1. 1l = 1.25y = 5y/4
  2. 2Cone total surface area = πy × 5y/4 + πy² = 5πy²/4 + πy² = 9πy²/4
  3. 34πR² = 9πy²/4 → y² = 16R²/9
  4. 4y = 4R/3

Answer4πR² = (5πy²)/4 + πy², completion to y = (4R)/3 with no errors

Part (b) · 4 marks

Using y = 4R/3, find the cone's volume in terms of R, simplified as far as possible.

  1. 1h² + y² = (5y/4)² → h² = 9y²/16 → h = 3y/4
  2. 2h = 3/4 × 4R/3 = R
  3. 3V = 1/3 × π × (4R/3)² × R = 1/3 × π × 16R²/9 × R
  4. 4= 16πR³/27

Answer(16πR³)/27

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Also searched as: 4024/11/O/N/22 · 4024_w22_qp_11 · 4024 w22 11 · question 23.
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