The answer
(a) \(\approx 95.6\) g (96 g to the nearest gram)
(b) shown
(c) open-ended, a worked, justified model below (e.g. small \(\approx\) $7, large \(\approx\) $18)
O-Level E-Math 2024 Paper 2 Question 9 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 9 of the O-Level E-Math 2024 Paper 2. It tests volume of a cylinder, in the Mensuration (similar solids) / Percentage / problem-solving area. It is worth 11 marks: 2 + 2 + 7. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) Volume \(= 5^3 = 125\) cm³. Total mass \(= 0.765 \times 125 = 95.625 \approx 95.6\) g (3 s.f.). (By the stated "nearest gram" rule this rounds to 96 g, see `_discrepancies.md`.)
(b) Let the mass of wax be \(x\) g. Total \(=\) wax \(+\) oil \(= x + 0.08x = 1.08x\). So \(1.08x = 150 \Rightarrow x = 138.9 \approx 139\) g (nearest gram). (shown)
(c) Open-ended; a defensible model. The cylinders are similar with diameters \(4:6 = 2:3\), so the small height \(8\) cm gives a large height \(12\) cm. Volumes: small \(= \pi(2^2)(8) = 32\pi = 100.5\) cm³; large \(= \pi(3^2)(12) = 108\pi = 339.3\) cm³ (volume ratio \(= (3/2)^3 = 27/8\)). Masses (rule): small \(= 0.765 \times 100.5 = 77\) g; large \(= 0.765 \times 339.3 = 260\) g. Splitting each as wax \(+\) 8% oil (\(\text{total} = 1.08 \times \text{wax}\)): small wax \(= 77/1.08 = 71\) g, oil \(\approx 6\) ml; large wax \(= 260/1.08 = 241\) g, oil \(\approx 19\) ml. Marginal material cost (using the cheaper unit rates: wax 1 kg \(= \$0.026\)/g, oil 60 ml \(= \$0.367\)/ml, wick \(\$8/10 = \$0.80\) each; each candle needs one 15 cm wick since the wick lengths \(8.5\) cm and \(12.5\) cm both fit): - Small: wax \(71(0.026)=\$1.85\) + oil \(6(0.367)=\$2.20\) + wick \(\$0.80 = \$4.85\). - Large: wax \(241(0.026)=\$6.27\) + oil \(19(0.367)=\$6.97\) + wick \(\$0.80 = \$14.04\).
Adding a small profit (about 30 to 40%) and rounding to a tidy price, Wei could charge about $7 for the small candle and $18 for the large candle. (Any answer is acceptable if the volumes, masses, costed materials and a stated profit margin are shown; pricing will vary with the assumptions made.)
Answer: (a) \(\approx 95.6\) g (96 g to the nearest gram)
(b) shown
(c) open-ended, a worked, justified model below (e.g. small \(\approx\) $7, large \(\approx\) $18)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a volume of a cylinder question from Mensuration (similar solids) / Percentage / problem-solving, worth 11 marks: 2 + 2 + 7.
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