4024/22 Oct/Nov 2021, Question 8: Volume and Surface Area of a Cone

Part (a) · 3 marks

Show the cone height h is 8.57 cm to 2 decimal places using the volume formula and diameter 7 cm.

1. 11/3 × π × 3.5² × h = 110
2. 2h = (3 × 110)/(π × 3.5²) = 330/38.48…
3. 3h = 8.574… = 8.57 to 2 d.p.

Answer(3 × 110)/(π × 3.5²) = 8.573 to 8.574…

Part (b) · 2 marks

Calculate the slant height l of the cup from its radius and height.

1. 1Radius = 7 ÷ 2 = 3.5 cm
2. 2110 = 1/3 × π × 3.5² × h, so h = 8.57… cm
3. 3l² = 3.5² + 8.57…² = 85.78…
4. 4l = √85.78… = 9.26 cm

Answer9.26 or 9.256 to 9.262

Part (c) · 4 marks

The cup is cut along OA and opened into a sector of radius 9.26 cm; calculate sector angle x.

1. 1Slant height OA = √(3.5² + 8.57²) = 9.26 cm (height 8.57 cm from part a)
2. 2Arc length of sector = circumference of cup top = π × 7
3. 3x/360 × 2 × π × 9.26 = π × 7
4. 4x = (360 × 7) ÷ (2 × 9.26) = 135.7 to 136.1…

Answer135.7 to 136.1…

Part (d) · 2 marks

A similar cup has volume 165 cm³; calculate the diameter of its top.

1. 1Volume scale factor = 165/110 = 1.5
2. 2Length scale factor = ∛1.5 = 1.1447…
3. 3Diameter = 7 × 1.1447… = 8.01… cm

Answer8.01...

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