The answer
(a) \(\approx 168\) cm²
(b)(i) volume scales with the cube of the depth, not linearly
(b)(ii) \(42.2\%\)
(b)(iii) \(r \approx 3.69\) cm
O-Level E-Math 2020 Paper 2 Question 5 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 5 of the O-Level E-Math 2020 Paper 2. It tests curved surface area of a cone, in the Mensuration / similar solids area. It is worth 10 marks: (a) 3, (b)(i) 1, (b)(ii) 2, (b)(iii) 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) Radius \(r = 5.5\), slant height \(l = \sqrt{5.5^2 + 8^2} = \sqrt{94.25} = 9.708\). Curved surface area \(= \pi r l = \pi (5.5)(9.708) = 167.7 \approx 168\) cm².
(b)(i) The water forms a smaller, similar cone of height \(6\) cm. The depth is \(\tfrac{6}{8} = 75\%\) of the full height, but that is a linear ratio; the volume ratio is \(\left(\tfrac{6}{8}\right)^3\), not \(\tfrac{6}{8}\), so the glass is not \(75\%\) full.
(ii) \(\left(\tfrac{6}{8}\right)^3 \times 100 = \left(\tfrac34\right)^3 \times 100 = \tfrac{27}{64} \times 100 = 42.2\%\).
(iii) Volume of water \(= \tfrac13 \pi (4.125)^2 (6) = 106.9\) cm³ (radius of the water cone \(= \tfrac{6}{8}\times 5.5 = 4.125\)). In the cylinder, \(\pi r^2 (2.5) = 106.9 \Rightarrow r^2 = 13.61 \Rightarrow r = 3.69\) cm.
Answer: (a) \(\approx 168\) cm²
(b)(i) volume scales with the cube of the depth, not linearly
(b)(ii) \(42.2\%\)
(b)(iii) \(r \approx 3.69\) 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 curved surface area of a cone question from Mensuration / similar solids, worth 10 marks: (a) 3, (b)(i) 1, (b)(ii) 2, (b)(iii) 4.
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