4024/22 May/Jun 2023, Question 8: Volume and Surface Area of a Cone

Part (ai) · 2 marks

Find the length and width of rectangle R, whose length is twice its width and perimeter is 20.4 cm.

1. 1Width w, length 2w: 2(2w + w) = 20.4
2. 26w = 20.4, w = 3.4
3. 3Length = 6.8 cm, width = 3.4 cm

Answerlength = 6.8, width = 3.4

Part (aii) · 2 marks

Rectangle S is similar to R and has perimeter 30.6 cm; calculate the length of S.

1. 1R: width w, length 2w, so perimeter = 6w = 20.4
2. 2w = 3.4, so length of R = 6.8 cm
3. 3Scale factor = 30.6 ÷ 20.4 = 1.5
4. 4Length of S = 1.5 × 6.8 = 10.2 cm

Answer10.2

Part (bi) · 2 marks

Find the arc length of a sector with angle 75° and radius 8 cm, in terms of π, simplified.

1. 1Arc length = 75/360 × 2 × π × 8
2. 2= 10π/3 cm

Answer10π/3 or 10/3 π or 3 1/3 π

Part (bii) · 5 marks

The sector forms the curved surface of a cone with slant height 8 cm; calculate the volume of water filling it.

1. 1Circumference of base = 10π/3, so 2πr = 10π/3 and r = 5/3
2. 2h = √(8² − (5/3)²) = √(64 − 25/9) = 7.824…
3. 3V = 1/3 × π × (5/3)² × 7.824…
4. 4V = 22.76… ≈ 22.8 cm³

Answer22.7 to 22.9

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