4024/22 May/Jun 2025, Question 21: Composite Perimeter, Area and Volume
Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 11 marks
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The question, in short
For a solid made of a cone joined to a hemisphere, both with diameter 14 cm and overall height 24 cm, work out its volume, verify its surface area, and use similarity to find a smaller solid's height.
Summarised in our own words. Use the official paper for the exact wording and the diagram.
Part (a) · 3 marks
Work out the volume of the combined cone and hemisphere solid using the given dimensions.
- 1Radius = 14 ÷ 2 = 7 cm; cone height = 24 − 7 = 17 cm
- 2Hemisphere = 1/2 × 4/3 × π × 7³ = 718.37…
- 3Cone = 1/3 × π × 7² × 17 = 872.31…
- 4Volume = 718.37… + 872.31… = 1590.69… ≈ 1590.7 cm³
Answer1590 or 1591 or 1590.6 to 1590.9
Part (b) · 5 marks
Demonstrate that the solid's total surface area rounds to 712 cm².
- 1Slant height = √(17² + 7²) = √338 = 18.38…
- 2Curved cone surface = π × 7 × 18.38… = 404.30…
- 3Hemisphere surface = 1/2 × 4 × π × 7² = 307.87…
- 4Total = 404.30… + 307.87… = 712.17…
- 5= 712 cm² to the nearest integer
Answer712.07 to 712.3
Part (c) · 3 marks
Given a similar solid with surface area 242 cm², work out its total height.
- 1Area scale factor = 242/712
- 2Length scale factor = √(242/712) = 0.5829…
- 3Height = 24 × 0.5829… = 13.99… ≈ 14.0 cm
Answer14.0 or 13.98 to 13.99…
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