O-LevelMathematicsMensuration4024/22 · Oct/Nov 2021 · Q811 Marks

[Volume of cone = (1/3)πr²h] [Curved surface area of a cone = πrl] The diagram shows a paper cup in the shape of a cone. The diameter of the top of the cup is 7 cm. The volume of the cup is 110 cm³. [Figure showing a cone with diameter 7 at the top, slant height l, and height h. NOT TO SCALE]

(a)Show that the height of the cup, h cm, is 8.57 correct to 2 decimal places.[3]
(b)Calculate the slant height, l cm, of the cup.[2]
(c)The cup is cut along the line OA. It is opened out into a sector of a circle with centre O and sector angle x°. Calculate the value of x.[4]
(d)A second paper cup is mathematically similar to the cup with volume 110 cm³. The volume of the second cup is 165 cm³. Calculate the diameter of the top of the second cup.[2]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

[M2] 3×110/(π×3.5²) oe [A1] = 8.573 to 8.574...
[2] M1 for 3.5² + 8.57²
[4] M3 for (360×π×7)/(2×π×their 9.26) oe or M2 for x/360 × 2×π×their 9.26 = π×7 oe or M1 for x/360 × 2×π×their 9.26 seen or π×7 oe seen Alternative method: M3 for (360×π×3.5×their 9.26)/(π×(their 9.26)²) oe or M2 for x/360 × π×(their 9.26)² = π×3.5×their 9.26 oe or M1 for x/360 × π×(their 9.26)² seen or π×3.5×their 9.26 seen
[2] M1 for ³√(165/110) × 7 oe or ³√(110/165) × diameter_of_second_cup = 7 oe

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