The answer
(a) shown
(b) \(d \approx 4.13\) cm
O-Level E-Math 2023 Paper 1 Question 18 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 18 of the O-Level E-Math 2023 Paper 1. It tests similar cones (linear relation of radius and height), in the Mensuration / Congruence & similarity area. It is worth 4 marks: 1 + 3. 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) The cone narrows linearly from radius 10 at the base to 0 at the apex over a height of 20. At height \(d\) above the base the radius is \(10 - \tfrac12 d\). The water surface there has radius \(r\), so \(r = 10 - \tfrac12 d \Rightarrow d = 20 - 2r\). (shown)
(b) The empty space is a small cone (apex at the top) similar to the whole cone, with base radius \(r\) and height \(h = 20 - d = 2r\). "Water \(=\) empty" means each is half the full cone: \[\tfrac13 \pi r^2 (2r) = \tfrac12 \times \tfrac13 \pi (10)^2 (20) \Rightarrow 2r^3 = \tfrac12 (2000) = 1000 \Rightarrow r^3 = 500 \Rightarrow r = 7.937.\] Then \(d = 20 - 2r = 20 - 15.874 = 4.13\) cm (3 s.f.).
Answer: (a) shown
(b) \(d \approx 4.13\) cm
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 similar cones (linear relation of radius and height) question from Mensuration / Congruence & similarity, worth 4 marks: 1 + 3.
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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