The answer
(a) \(h = 60\)
(b) \(d \approx 87.4\) cm
O-Level E-Math 2024 Paper 1 Question 26 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 26 of the O-Level E-Math 2024 Paper 1. It tests volume of a cylinder & cone, in the Mensuration area. It is worth 6 marks: (a) 2, (b) 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) The water exactly fills the cylinder, and this is half the container's volume. So the cylinder's volume equals the cone's volume (each is half the total). With equal radii, \(\pi r^2 (20) = \tfrac{1}{3}\pi r^2 h \Rightarrow 20 = \tfrac{h}{3} \Rightarrow h = 60\).
(b) The two solids have radius \(r = 10\) cm. \[V_{\text{cylinder}} = \pi(10)^2(25) = 2500\pi = 7853.98\text{ cm}^3, \qquad V_{\text{cone}} = \tfrac{1}{3}\pi(10)^2(90) = 3000\pi = 9424.78\text{ cm}^3.\] Total volume \(= 17278.76\) cm³, so half (the water) \(= 8639.38\) cm³, and the empty space \(= 8639.38\) cm³. The empty region is the top part of the cone, a smaller cone similar to the whole cone, with height \(d\) and apex at the top. For similar cones the volume ratio is the cube of the height ratio: \[\left(\dfrac{d}{90}\right)^3 = \dfrac{V_{\text{empty}}}{V_{\text{cone}}} = \dfrac{8639.38}{9424.78} = 0.91667 \;\Rightarrow\; \dfrac{d}{90} = \sqrt[3]{0.91667} = 0.9712 \;\Rightarrow\; d = 87.4\text{ cm (3 s.f.)}.\]
Answer: (a) \(h = 60\)
(b) \(d \approx 87.4\) 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 volume of a cylinder & cone question from Mensuration, worth 6 marks: (a) 2, (b) 4.
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