O-LevelMathematicsMensuration4024/22 · Oct/Nov 2022 · Q99 Marks

[Volume of a sphere = 4/3 πr³] [Surface area of a sphere = 4πr²] The diagram shows a wooden bowl. It is made in the shape of a large hemisphere with a small hemisphere removed from the centre. The diameter of the large hemisphere is 20 cm. The width of the rim of the bowl is 2 cm. [Figure of a cross-section of a bowl, showing outer radius, inner radius, and rim width]

(a)Show that the total surface area of the bowl is 364π cm².[3]
(b)The bowl is made from wood. The mass of 1 cm³ of the wood is 0.74 g. Calculate the mass of the bowl.[3]
(c)Another bowl is mathematically similar to the first bowl and is made from the same type of wood. The total surface area of the second bowl is 546π cm². Calculate the mass of the second bowl.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[3] M2 for 2π×10² + 2π×8² + π×10² – π×8² or M1 for 2π×10² seen or 2π×8² seen or π×10² – π×8² seen A1 for Completion to 364π with at least one intermediate step isw AG A0 if any errors or if π evaluated as 3.14[2...] or 22/7 before getting to 364π
[3] M2 for 2/3π×10³ – 2/3π×8³ ×0.74 oe or M1 for 2/3π×10³ – 2/3π×8³ oe or 2/3π×k³ [+...]×0.74 oe with k > 0
[3] M2 for (√(546/364))³ oe soi or (√(546/364))³ oe soi or M1 for √(546/364) oe soi or √(546/364) oe soi or (364^(1/3))/(546^(1/3)) = (their b)^(1/2) / V^(1/2) oe

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