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4024/22 May/Jun 2024, Question 8: Composite Perimeter, Area and Volume

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 15 marks

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The question, in short

A cone sits on a hemisphere with equal radius r and total height 10 cm; hemisphere curved area is 145 cm². Find r, the solid's volume, and the cone's curved area; then compare two sectors' areas.

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

Part (ai) · 3 marks

Using the hemisphere's curved area of 145 cm², show that the radius works out to 4.80 cm when rounded to two decimal places.

  1. 1Curved surface of hemisphere = 1/2 × 4πr² = 2πr²
  2. 22πr² = 145, so r² = 145/(2π) = 23.07…
  3. 3r = 4.803… = 4.80 (2 d.p.)

Answerr = 4.803 to 4.804

Part (aii) · 4 marks

Work out the total volume of the cone-plus-hemisphere solid, using the radius found.

  1. 1Height of cone = 10 − 4.8 = 5.2 cm
  2. 2Cone: 1/3 × π × 4.8² × 5.2 = 125.4…
  3. 3Hemisphere: 1/2 × 4/3 × π × 4.8³ = 231.6…
  4. 4Volume = 357.08… ≈ 357 cm³

Answer357 to 358

Part (aiii) · 4 marks

Work out the curved surface area of the cone part of the solid.

  1. 1Slant height l = √(4.8² + 5.2²) = √50.08 = 7.076…
  2. 2Curved surface area = π × 4.8 × 7.076…
  3. 3= 106.7… ≈ 107 cm²

Answer107 or 106.7 to 106.8…

Part (b) · 4 marks

Sector B's angle is 20% less and its radius 20% more than sector A's; find B's area as a percentage of A's area.

  1. 1Angle of B = 0.8x, radius of B = 1.2y
  2. 2Area B = 0.8x/360 × π(1.2y)² = 0.8 × 1.44 × (x/360 × πy²)
  3. 30.8 × 1.2² = 1.152
  4. 4Area B = 115.2% of area A

Answer115.2

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Also searched as: 4024/22/M/J/24 · 4024_s24_qp_22 · 4024 s24 22 · question 8.
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