4024/21 Oct/Nov 2024, Question 6: Surface Area of a Cuboid and Cylinder
Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 10 marks
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The question, in short
Three-part geometry problem: find a sphere's volume inside a cube of volume 343 cm³, find a similar solid's height from volumes, and find a cone-on-cylinder solid's total surface area.
Summarised in our own words. Use the official paper for the exact wording and the diagram.
Part (a) · 3 marks
Cube volume 343 cm³ with a sphere touching all six faces; calculate the sphere's volume using 4/3 πr³.
- 1Cube side = ∛343 = 7 cm, so sphere radius = 3.5 cm
- 2Volume = 4/3 × π × 3.5³
- 3= 179.59… ≈ 180 cm³
Answer180 or 179.5 to 179.6…
Part (b) · 2 marks
Similar solids A and B have volumes 540 cm³ and 1280 cm³; A's height is 15 cm, find B's height.
- 1Volume scale factor = 1280/540 = 64/27
- 2Length scale factor = ∛(64/27) = 4/3
- 3Height of B = 15 × 4/3 = 20 cm
Answer20
Part (c) · 5 marks
Cone radius 6.3 cm, slant height 8.7 cm, cone:cylinder height ratio 2:3; find total surface area of the joined solid.
- 1Cone height = √(8.7² − 6.3²) = √36 = 6 cm
- 2Cylinder height = 6 ÷ 2 × 3 = 9 cm
- 3Cone curved surface = π × 6.3 × 8.7; cylinder curved surface = 2π × 6.3 × 9; base = π × 6.3²
- 4Total = π(54.81 + 113.4 + 39.69) = 207.9π
- 5= 653.13… ≈ 653 cm²
Answer653 or 653.1 to 653.2…
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