0580/41 Oct/Nov 2023, Question 8: Composite Perimeter, Area and Volume

Part (ai) · 4 marks

Show the total volume of the solid, using radii 5.4 cm and 3.6 cm and cylinder height 6.5 cm, equals 692 cm³ to the nearest cm³.

1. 1Small hemisphere = 2/3 × π × 3.6³ = 97.71…
2. 2Cylinder = π × 3.6² × 6.5 = 264.64…
3. 3Large hemisphere = 2/3 × π × 5.4³ = 329.79…
4. 4Total = 692.15… ≈ 692 cm³

Answer692.1 to 692.2…

Part (aii) · 4 marks

For a similar solid with cylinder radius 0.6 cm, use the scale factor and density 10.49 g/cm³ to find the total mass.

1. 1Length scale factor = 0.6/3.6 = 1/6
2. 2Volume = 692 × (1/6)³ = 3.2037… cm³
3. 3Mass = 3.2037… × 10.49 = 33.60… ≈ 33.6 g

Answer33.6 or 33.60 to 33.62

Part (bi) · 2 marks

Calculate the arc length of a sector with radius 10 cm and angle 216°, giving the answer as a multiple of π.

1. 1Arc = 216/360 × 2 × π × 10
2. 2= 12π cm

Answer12π

Part (bii) · 4 marks

The sector is formed into a cone by joining OA to OB; calculate the cone's volume using its radius and height.

1. 1Arc length = 216/360 × 2 × π × 10 = 12π cm
2. 2The arc becomes the base circumference: 2πr = 12π, so r = 6 cm.
3. 3Slant height is 10 cm, so h = √(10² − 6²) = 8 cm.
4. 4V = 1/3 × π × 6² × 8 = 96π = 301.59… ≈ 302 cm³

Answer302 or 301.5 to 301.6…

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