The answer
(a) \(\approx 226 \text{ cm}^3\)
(b) shown
(c) \(\approx 1240 \text{ cm}^3\)
O-Level E-Math 2015 Paper 2 Question 7 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 7 of the O-Level E-Math 2015 Paper 2. It tests height of a pyramid via 3d pythagoras, in the Mensuration (pyramid, sphere, cuboid) area. It is worth 9 marks: (a) 4, (b) 2, (c) 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 apex \(E\) is above the centre of the base. The base diagonal is \(8\sqrt2\), so the distance from the centre to a corner is \(\dfrac{8\sqrt2}{2} = 4\sqrt2\). Height \(h = \sqrt{12^2 - (4\sqrt2)^2} = \sqrt{144 - 32} = \sqrt{112} = 10.583\) cm. Volume \(= \dfrac13 \times 8^2 \times 10.583 = \dfrac13 \times 64 \times 10.583 = 225.8 \approx 226 \text{ cm}^3\) (3 s.f.).
(b) Sphere volume \(=\) candle volume: \(\dfrac43 \pi r^3 = 225.8 \Rightarrow r^3 = \dfrac{3 \times 225.8}{4\pi} = 53.9 \Rightarrow r = \sqrt[3]{53.9} = 3.78\) cm (3 s.f.). (shown)
(c) The six spheres (diameter \(2 \times 3.78 = 7.56\) cm) fit a \(3 \times 2 \times 1\) arrangement, so the box is \((3 \times 7.56) \times (2 \times 7.56) \times 7.56 = 22.68 \times 15.12 \times 7.56 = 2592 \text{ cm}^3\). The six candles occupy \(6 \times 225.8 = 1355 \text{ cm}^3\). Empty space \(= 2592 - 1355 = 1237 \approx 1240 \text{ cm}^3\) (3 s.f.).
Answer: (a) \(\approx 226 \text{ cm}^3\)
(b) shown
(c) \(\approx 1240 \text{ cm}^3\)
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 height of a pyramid via 3d pythagoras question from Mensuration (pyramid, sphere, cuboid), worth 9 marks: (a) 4, (b) 2, (c) 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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