The answer
(a) shown (\(27\pi y^2\))
(b) \(h = \dfrac{19y}{4}\)
(c) \(\approx 528\) cm³
O-Level E-Math 2023 Paper 2 Question 3 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 3 of the O-Level E-Math 2023 Paper 2. It tests surface area of sphere/hemisphere, in the Mensuration area. It is worth 9 marks: 2 + 3 + 4. 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) A solid hemisphere has a curved surface \(2\pi r^2\) and a flat circular face \(\pi r^2\), so total surface area \(= 3\pi r^2\). With \(r = 3y\): \(3\pi(3y)^2 = 3\pi \cdot 9y^2 = 27\pi y^2\) cm². (shown)
(b) Cylinder TSA \(= 2\pi r h + 2\pi r^2\) with \(r = 2y\): \(= 2\pi(2y)h + 2\pi(2y)^2 = 4\pi y h + 8\pi y^2\). Set equal to the hemisphere's \(27\pi y^2\): \(4\pi y h + 8\pi y^2 = 27\pi y^2 \Rightarrow 4\pi y h = 19\pi y^2 \Rightarrow h = \dfrac{19y}{4}\).
(c) Compare volumes directly. Hemisphere volume \(= \tfrac{2}{3}\pi(3y)^3 = \tfrac{2}{3}\pi \cdot 27y^3 = 18\pi y^3\). Cylinder volume \(= \pi(2y)^2 h = \pi \cdot 4y^2 \cdot \tfrac{19y}{4} = 19\pi y^3\). So \(\dfrac{\text{cylinder}}{\text{hemisphere}} = \dfrac{19}{18}\), giving cylinder volume \(= \dfrac{19}{18} \times 500 = 527.8 \approx 528\) cm³.
Answer: (a) shown (\(27\pi y^2\))
(b) \(h = \dfrac{19y}{4}\)
(c) \(\approx 528\) cm³
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 surface area of sphere/hemisphere question from Mensuration, worth 9 marks: 2 + 3 + 4.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
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